https://dev.imagemakers.co.za/guides/solution-we-are-given-a-multiset-of-10-components-3-identical-sensors-s-5-identical-drones-d-and-2-identical-robotic-arms-r-the-number-of-distinct-activation-sequences-is-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-sequences-is-given-by-the-multinomial-coefficient-10534112026-02-18T04:29:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multiset-permutation-problem-with-7-drones-3-identical-multispectral-m-2-thermal-t-and-2-lidar-l-the-number-of-distinct-sequences-is-44906492026-02-18T04:29:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-shelves-be--n--the-number-of-panels-forms-an-arithmetic-sequence-7-11-15--with-first-term--a--7--and-common-difference--d--4--the-sum-of-the-first--n--terms-is-36313982026-02-18T04:29:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-population-grows-as--324-times-2t3--where--t--is-time-in-years-we-want-the-smallest--k--such-that-33280252026-02-18T04:29:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-two-vectors-to-be-orthogonal-their-dot-product-must-be-zero-compute-the-dot-product-42424862026-02-18T06:54:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-fourth-vertex-be-d--x-y-z-in-a-regular-tetrahedron-all-six-edges-must-be-equal-first-compute-the-distance-between-known-points-34935622026-02-18T06:54:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-how-many-combinations-of-3-flavors-and-1-topping-a-customer-can-choose-we-calculate-the-number-of-ways-to-select-3-flavors-from-8-and-1-topping-from-5-92685602026-02-18T08:49:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-binomial-probability-problem-with--n--4--trials-rolls-success-probability--p--frac16--rolling-a-4-and-we-want-exactly--k--2--successes-12329862026-02-18T08:49:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-identify-the-vowels-and-consonants-in-bioinformatics-7974202026-02-18T08:49:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-5-distinct-integers-independently-and-uniformly-at-random-from--0-1-2-ldots-50--and-let--z--be-the-sum-of-the-two-largest-we-want--mathbbpz-geq-100--88119062026-02-18T08:49:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-integers--n--such-that--n-equiv-2-pmod5--among-the-first-100-positive-integers-43836982026-04-20T16:46:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--69532382026-04-20T16:46:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-centroid--g--of-triangle--abc--is-the-average-of-its-vertices-coordinates-6905482026-04-20T16:55:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-center-of-mass-centroid-of-triangle--abc--2063812026-04-20T16:55:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--fx--ax3--bx2--cx--d--and-the-values-19863842026-04-20T16:55:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-slope-of-the-terrain-is-given-by-the-derivative-26542552026-04-20T16:55:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-rectangle-is-inscribed-in-the-circle-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-given-by-72042112026-04-20T17:15:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-s-is-61046532026-04-20T17:15:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-perimeter-of-the-original-triangle-is-48115662026-04-20T17:15:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-identity-74145182026-04-20T17:15:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-circumradius-r-of-a-triangle-985502026-04-20T17:15:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-circle-is-given-by--a--pi-r2--setting--pi-r2--25pi--divide-both-sides-by--pi--to-get--r2--25--taking-the-square-root-yields--r--5--boxed5-96713862026-04-20T17:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-maximum-value-of--sin--is-1-so-the-maximum-depth-occurs-when--sinleftfracpi-t12right--1--substituting--dtextmax--10--41--14--boxed14-3748942026-04-20T17:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate--g2--22---1--4---1--3--then-substitute-into--fx---f3--23--3--6--3--9--boxed9-70674822026-04-20T17:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-from--x---y--1--express--x--y--1--substitute-into-the-first-equation--3y--1--2y--6-rightarrow-3y--3--2y--6-rightarrow-5y--3-rightarrow-y--frac35--then--x--frac35--1--frac85--boxeddfrac85-55786032026-04-20T17:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--n--3---m3--32---43--7--9---12--7--4--boxed41-a-train-travels-150-miles-in-3-hours-at-a-constant-speed-if-it-continues-at-the-same-speed-how-many-miles-will-it-cover-in-5-hours-18347872026-04-20T17:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--3218--to-base-ten-we-expand-it-using-powers-of-8-40892332026-04-20T17:18:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor--180--22-cdot-32-cdot-5--to-find-only-the-odd-divisors-we-ignore-the-factor-of-2-and-consider-only--32-cdot-5--69689092026-04-20T17:18:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-two-digit-number-we-are-given-99828562026-04-20T17:18:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--their-product-is--nn1n2n3--30262522026-04-20T17:18:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-is-the-perpendicular-projection-the-slope-of-the-line-is--m---frac12--so-the-perpendicular-slope-is--2--the-equation-of-the-perpendicular-line-through--4-3--is--y---3--2x---4--solving-the-system-74164482026-04-20T17:36:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-columns-of--mathbfm--are-the-images-of-the-standard-basis-vectors-thus--mathbfm--beginbmatrix-3--1---2--4-endbmatrix--verification-2995432026-04-20T17:36:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-de-moivres-theorem--cos-theta--i-sin-thetan--cosntheta--i-sinntheta--here--n--5---theta--72circ--21470262026-04-20T17:36:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-using-the-identity--sin2-x--1---cos2-x--to-rewrite--fx--3781182026-04-20T17:36:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite--tan-theta--cot-theta--4--as-82609782026-04-20T17:36:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--vecoa-cdot-vecob--0--they-are-perpendicular-let--vecoa--3-0---vecob--0-4--then--vecog--3m-4n--40642592026-04-20T17:36:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfv--langle-v1-v2-v3-rangle-the-cross-product-mathbfv-times-mathbfa--mathbfb-gives-85053752026-04-20T17:38:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-u--sin-z-the-equation-becomes-2u2---3u--1--0-solving-93037612026-04-22T05:55:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--x2--y2--25--represents-a-circle-of-radius-5-centered-at-the-origin-we-seek-all-integer-pairs--x-y--that-satisfy-this-the-integer-solutions-occur-when--x2--and--y2--are-perfect-squares-summing-to-25-the-possible-square-pairs-are-71463722026-04-20T17:40:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-maximum-height-of-a-parabola--y--ax2--bx--c--occurs-at--x---fracb2a--here--a---1---b--6--so--x---frac62-1--3--substituting--x--3--into-the-equation-12402622026-04-20T17:40:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-the-logarithmic-equation-in-exponential-form--x--3--24--simplify--24--16--so--x--16---3--13--87932542026-04-20T17:40:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-number-of-successful-tasks-in-the-additional-10-attempts-initially-successful-tasks---085-imes-40--34--total-tasks-after-10-more---50--total-successful-tasks---34--x--new-success-rate--rac34--x50--088--solving--34--x--44--so--x--10--44863082026-04-20T18:02:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--a--5--into--55--b--120--simplify--25--5b--120--subtract-25--5b--95--divide-by-5--b--19--62796122026-04-20T18:02:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-in-slope-intercept-form--3y--2x--12----y--rac23x--4--the--y--intercept-occurs-at--x--0--so--y--4--67106242026-04-20T18:02:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve--x--3---rac12--rac52--substitute-into--4x--2---4-imes-rac52--2--10--2--12--69110242026-04-20T18:02:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--r4--2r0--substitute-into-the-equation--1000e4k--2-cdot-1000--simplify-to--e4k--2--take-the-natural-logarithm--4k--ln2--so--k--racln24--thus--oxeddfracln24--36062512026-04-20T18:02:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify-the-left-side--racx-3x3x-3--x--3--provided--x-3729712026-04-20T18:02:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides--a--5--49--subtract-5--a--44--verify-by-substituting-back--sqrt44--5--sqrt49--7--final-answer--oxed44--85444642026-04-20T18:02:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-prime-factorization-of-16-is-24-and-of-24-is-23-imes-3-the-lcm-is-the-product-of-the-highest-powers-24-imes-3--16-imes-3--48-thus-the-lcm-is-oxed48-20874022026-04-20T18:02:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-x-equiv-3-pmod7-and-x-equiv-5-pmod9-let-x--7k--3-substitute-into-the-second-congruence-7k--3-equiv-5-pmod9-rightarrow-7k-equiv-2-pmod9-the-modular-inverse-of-7-modulo-9-is-4-since-7-imes-4--28-equiv-1-pmod9-so-k-equiv-2-imes-4--8-pmod9-thus-k--9m--8-and-x--79m--8--3--63m--59-the-solutions-are-59-122-185-dots-within-1150-only-59-and-122-satisfy-the-count-is-oxed2-42711812026-04-20T18:02:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-lcm-of-13-and-17-is-223-since-223--200-the-largest-multiple-is-0-invalid-adjusting-the-largest-under-200-is-0-no-valid-number-however-if-the-question-allows-divisibility-by-13-or-17-the-answer-differs-assuming-exact-lcm-the-answer-is-oxed0-no-solution-41561322026-04-20T18:02:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-sequence-have-first-term--a1--15--and-fifth-term--a5--35--the-common-difference--d--satisfies--a5--a1--4d--substituting--35--15--4d-rightarrow-d--5--the-sum-of-the-first--n--10--terms-is--s10--rac102-215--95--530--45--575--375--oxed375-59364342026-04-20T18:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-points-per-correct-challenge-for-10-challenges-there-are--lfloor-rac10---13-17686462026-04-20T18:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-intervals-is--rac1815--12--since-beams-are-placed-at-both-ends-total-beams---12--1--13--however-the-problem-states-4-beams-per-3-meters-which-implies--rac43--beams-per-meter-for-18-meters--rac43-imes-18--24--oxed24-47396022026-04-20T18:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-score-at-level--n--be--sn--s1-cdot-2n-1--given--s3--s1-cdot-4--128-rightarrow-s1--32--then--s5--32-cdot-24--32-cdot-16--512--oxed512-89501022026-04-20T18:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with--a1--8--and--d--2--the-7th-term-is--a7--a1--6d--8--12--20--oxed20-37871002026-04-20T18:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio--rac5-ext-bolts2-ext-beams--implies--rac30-ext-boltsx-ext-beams--rac52--cross-multiplying--5x--60-rightarrow-x--12--oxed121-75097312026-04-20T18:19:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initially-the-explorer-has--25-times-060--15--successful-voyages-after-10-more-attempts-the-total-voyages-are--25--10--35--with-a-68-success-rate-total-successful-voyages-become--35-times-068--238--which-rounds-to-24-since-successes-are-whole-numbers-the-additional-successful-voyages-are--24---15--9--76342072026-04-20T18:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-quadratic--pt---t2--14t--30--opens-downward-the-vertex-occurs-at--t---fracb2a---frac142-1--7--substituting--t--7---p7---72--147--30---49--98--30--79--72888572026-02-23T03:57:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--y--4x---5--into-the-second-equation--24x---5--3x--10--simplify--8x---10--3x--10-rightarrow-11x--20-rightarrow-x--frac2011--substitute-back--y--4leftfrac2011right---5--frac8011---frac5511--frac2511--72059382026-04-20T18:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-distributive-property-98099732026-04-20T18:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-48-and-60-factor-both-numbers-35828862026-04-20T18:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-8-nucleotide-sequences-using-exactly-2-as-3-cs-and-3-gs-total-8-nucleotides-such-that-no-two-identical-nucleotides-are-adjacent-77147932026-04-20T18:46:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-unique-2-nucleotide-combinations-from-5-distinct-nucleotides-without-considering-order-we-use-the-combination-formula--binomnr--where--n--5--and--r--2--60548432026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-represent-the-arrival-times-as-points-x-y-in-a-60-minute-square-the-condition--y--x--15--order-matters-corresponds-to-a-triangular-region-with-area--frac12-times-60---152--frac12-times-2025--10125--the-total-area-is--602--3600--the-probability-is--frac101253600--frac932--18525462026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-requires-the-stirling-numbers-of-the-second-kind--s5-3--which-count-the-ways-to-partition-5-distinguishable-objects-into-3-non-empty-indistinct-subsets-using-the-recurrence-relation-35417522026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-3-genes-from-8-is-given-by-the-combination-formula-64291242026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-represent-the-times-as-x-y-in-a-60-minute-square-the-condition--y-geq-x--20--defines-a-triangular-region-with-base--60---20--40--minutes-the-area-of-this-region-is--frac12-times-402--800--the-total-area-is--602--3600--the-probability-is--frac8003600--frac29--94390182026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-partitioning-4-distinguishable-objects-into-2-non-empty-indistinct-subsets-calculated-by-the-stirling-number--s4-2--using-the-recurrence-55364312026-04-20T19:39:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--sqrtx--so--u-geq-0--substituting-the-equation-becomes--u--3u2--4--rearranging--3u2--u---4--0--using-the-quadratic-formula--u--frac-1-pm-sqrt1--486--frac-1-pm-76--valid-solution--u--1--since--u-geq-0--thus--sqrtx--1-rightarrow-x--1--boxed1-10845392026-04-20T19:41:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--2y--1-rightarrow-y--fract---12--substitute-into--ht---ht--leftfract---12right3---5leftfract---12right--6--simplify--ht--fract-138---frac5t-12--6--now-replace--t--with--y2---1---hy2---1--fracy2---238---frac5y2---22--6--expand-and-combine-terms-if-required-boxedfracy2---238---frac5y2---22--6-52756772026-04-20T19:41:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-group-and-complete-the-square--4x2--2x---9y2---2y--11--complete-squares--4x12---1---9y-12---1--11--expand--4x12---4---9y-12--9--11-rightarrow-4x12---9y-12--6--rewrite--fracx12frac32---fracy-12frac23--1--center-at---1-1--boxed-1-11-a-chemistry-experiment-requires-50-ml-of-a-10-salt-solution-if-you-only-have-a-20-salt-solution-and-distilled-water-how-much-of-each-should-be-mixed-to-obtain-the-desired-concentration-71314612026-02-23T05:53:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-the-number-of-distinct-ways-to-assign-10-independently-classified-images-into-4-categories-with-fixed-counts-3-as-tumor-2-as-inflammation-4-as-normal-and-1-as-stroke-this-is-a-multinomial-coefficient-problem-94167512026-04-20T19:44:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-distinct-arrangements-of-banana-without-restrictions-the-word-has-6-letters-with-repetitions-3-as-2-ns-and-1-b-22490142026-04-20T19:44:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-continuous-probability-problem-involving-uniform-random-variables-and-coverage-intervals-81307512026-04-20T19:44:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-greatest-common-divisor-of--5m---1--and--5n---1--for-positive-integers--m--and--n--is-given-by-the-identity-76815702026-04-20T19:44:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-sqrt52--82--sqrt25--64--sqrt89-therefore-the-radius-of-the-circle-is-fracsqrt892-the-circumference-is-given-by-2pi-r--2pi-cdot-fracsqrt892--pisqrt89-25379312026-04-20T20:02:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-is-frac43pi-2x3--frac43pi-8x3--frac323pi-x3-the-volume-of-a-hemisphere-is-frac23pi-3x3--frac23pi-27x3--18pi-x3-the-ratio-of-the-volumes-is-3479612026-04-20T20:02:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-88351612026-04-20T20:02:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-the-first--n--cubes-is-given-by-the-formula-95408102026-04-20T20:37:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-11071-11073-11075-11077-11079-the-sum-of-an-arithmetic-sequence-is-given-by-s--fracn2a1--an-here-n--5-a1--11071-and-a5--11079-65817372026-04-20T21:08:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-x2---y2--2025-factors-as-x---yx--y--2025-since-x-and-y-are-integers-both-x---y-and-x--y-must-be-integers-let-a--x---y-and-b--x--y-so-ab--2025-then-x--fraca--b2-and-y--fracb---a2-for-x-and-y-to-be-integers-a--b-and-b---a-must-both-be-even-so-a-and-b-must-have-the-same-parity-68710362026-04-20T21:08:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-odd-integers-be-n-n2-n4-n6-where-n-is-odd-their-product-is-72964492026-04-20T21:08:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs--a--and--b--and-hypotenuse--z--the-inradius-is-given-by--c--raca--b---z2--the-area--a--of-the-triangle-is--rac12ab--the-area-of-the-incircle-is--pi-c2--using-the-identity--a2--b2--z2--and-expressing--a--b--in-terms-of--z--and--c--we-find--a--b--z--2c--also-the-semiperimeter--s--raca--b--z2--rac2z--2c2--z--c--the-area-can-also-be-written-as--a--r-cdot-s--cz--c--therefore-the-ratio-becomes-34433592026-04-20T21:21:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-spherical-cap-of-height--h--cut-from-a-sphere-of-radius--r--is-given-by-the-formula-77901032026-04-20T21:21:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-of-radius--x--is--vextparticle--rac43pi-x3--the-volume-of-a-hemisphere-of-radius--3x--is-half-the-volume-of-a-full-sphere-91995712026-04-20T21:21:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle--120circ--lies-in-the-second-quadrant-using-the-identity--an180circ---heta---an-heta--we-write-55556922026-04-20T21:21:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-12-even-numbers-after-2020-are-2022-2024-ldots-2044-the-sum-of-an-arithmetic-sequence-is-s--racn2a1--an-here-n--12-a1--2022-an--2044-thus-s--rac1222022--2044--6-imes-4066--24396-dividing-24396-by-11-11-imes-2216--24376-so-the-remainder-is-24396---24376--20-however-20-mod-11--9-therefore-the-remainder-is-oxed9-14694742026-04-20T21:21:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-prime-factorization-of-2048-is-211-and-1024-is-210-the-lcm-of-two-numbers-is-the-product-of-the-highest-powers-of-all-primes-present-thus-extlcm2048-1024--211--2048-the-answer-is-oxed2048-17913292026-04-20T21:21:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-numbers-congruent-to-3-mod-7-are-3-10-17-ldots-3--7k-the-largest-such-number-leq-100-satisfies-3--7k-leq-100-rightarrow-k-leq-rac977-pprox-13857-thus-k--0-1-ldots-13-giving-14-numbers-the-count-is-oxed14-89050562026-04-20T21:21:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-prime-factorization-of-2025-is-34-imes-52-and-1024-is-210-since-there-are-no-common-prime-factors-the-gcd-is-1-the-answer-is-oxed1-69489512026-04-20T21:21:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-squares-formula-gives-s--rac15-imes-16-imes-316--1240-compute-1240-mod-13-13-imes-95--1235-so-1240---1235--5-the-remainder-is-oxed5question-a-science-educator-is-designing-a-lesson-on-probability-using-a-digital-simulation-that-generates-random-6-digit-numbers-using-only-the-digits-1-2-and-3-what-is-the-probability-that-a-randomly-generated-number-has-exactly-two-consecutive-digits-that-are-the-same-30699732026-04-20T21:22:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-must-count-the-number-of-6-digit-numbers-using-only-the-digits-1-2-and-3-such-that-exactly-one-pair-of-consecutive-digits-are-equal-and-all-other-adjacent-digits-are-different-91512072026-04-20T21:21:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-probability-that-the-product-of-three-randomly-chosen-integers-between-0-and-100-is-divisible-by-5-we-use-complementary-probability-first-note-that-an-integer-is-divisible-by-5-if-it-ends-in-0-or-5-among-the-integers-from-0-to-100-there-are-21-multiples-of-5-ie-0-5-10--100-thus-the-probability-that-a-single-randomly-chosen-integer-is-not-divisible-by-5-is-31448092026-04-20T21:35:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-determine-how-many-years-between-1800-and-1900-inclusive-are-divisible-by-4-17791772026-04-20T21:35:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-papers-from-12-is-17536982026-04-20T21:35:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-sites-from-10-is-7330822026-04-20T21:35:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-sequence-is-an-arithmetic-sequence-with-the-first-term--a--3--and-common-difference--d--5--the-general-term-of-the-sequence-is-given-by-95832442026-04-20T22:27:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-8-and-12-we-first-determine-their-prime-factorizations-98016772026-04-20T22:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-prime-factor-of-77-we-test-divisibility-by-the-smallest-prime-numbers-32702212026-04-20T22:27:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-value-of-sec-theta--csc-theta2-first-express-it-in-terms-of-sine-and-cosine-77816432026-04-20T22:27:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-95624332026-04-20T22:29:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-point-on-the-line-y--2x--3-closest-to-1--1-we-minimize-the-distance-between-x-2x--3-and-1--1-the-squared-distance-is-24304762026-02-23T10:55:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-y--rac2sin-x--3sin-x--2-solve-for-sin-x-31422612026-04-20T22:54:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfm--eginbmatrix-a--b--c--d-endbmatrix-25418892026-04-20T22:54:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-expand-9684392026-04-20T22:54:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-points-lie-on-the-plane-z--3-the-vector-from-1-2-3-to-1-5-3-is-0-3-0-and-the-vector-to-4-2-3-is-3-0-0-the-fourth-vertex-on-the-same-face-is-obtained-by-adding-these-vectors-1--3-2--3-3--4-5-3-49371482026-04-20T23:03:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product-3x--2-1---14--3x---2---4--3x---6-set-this-equal-to-zero-3x---6--0-implies-x--2-66107702026-04-20T23:03:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-cross-product-mathbfv-times-mathbfa-is-computed-as-92842192026-04-20T23:03:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-sin2x--frac12-has-solutions-when-2x--fracpi6--2pi-n-or-2x--frac5pi6--2pi-n-for-integer-n-solving-for-x-in-0-2pi-57790222026-04-20T23:04:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-dot-product-mathbfu-cdot-mathbfv--mathbfw--mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw-since-mathbfu-mathbfv-and-mathbfw-are-unit-vectors-each-dot-product-is-at-most-1-the-maximum-occurs-when-mathbfu-aligns-with-both-mathbfv-and-mathbfw-ie-mathbfv--mathbfw--mathbfu-then-mathbfu-cdot-mathbfv--mathbfw--1--1--2-11869192026-04-20T23:04:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-5-distinct-parts-a-b-c-d-and-e-and-we-are-to-count-the-number-of-permutations-where-a-appears-before-b-and-c-appears-before-d-64149502026-04-20T23:07:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-8-segment-sequences-using-letters-a-u-c-g-such-that-90320392026-04-20T23:07:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-probability-that-a-single-project-is-exceptionally-strong-ie-gets-h-in-at-least-5-of-the-3-categories-but-there-are-only-3-categories-so-getting-high-in-at-least-5-is-impossible-66042792026-04-20T23:07:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-robot-takes-8-steps-each-independently-choosing-to-go-right-increase-x-or-up-increase-y-uniformly-at-random-assume-fair-walk-the-endpoint-is-x-8---x-for-x--01dots8-so-x-follows-a-binomial-distribution-x-sim-textbinomial8-frac12-13588292026-04-20T23:07:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-4-protein-combinations-from-6-proteins--binom64--15--99798152026-04-20T23:07:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-ranking-configurations-of-5-projects-where-265672026-04-20T23:07:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-smallest-positive-angle-at-which-the-robot-returns-to-its-original-orientation-we-must-find-the-least-common-multiple-lcm-of-the-two-rotation-increments-12circ-and-18circ-first-factor-both-numbers-20509702026-04-20T23:34:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-city-is-2025-km-and-we-are-to-find-how-many-integer-side-lengths-s-correspond-to-square-grid-sizes-such-that-s2-divides-2025-this-requires-finding-the-number-of-positive-divisors-of-2025-since-each-such-divisor-s2-implies-s-is-an-integer-divisor-of-sqrt2025--45-29481912026-04-20T23:34:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-make-the-substitution--x--sqrtu--so-that--u--x2--substituting-into-the-equation-gives-20776652026-04-20T23:35:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-91440972026-04-20T23:35:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs--a--and--b--and-hypotenuse--c--the-radius--r--of-the-inscribed-circle-is-given-by-49678872026-04-20T23:48:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-square-is-the-diameter-of-the-circle-for-a-square-with-side-length--s--the-diagonal--d--is-67779872026-04-20T23:48:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s--is-67099762026-04-20T23:48:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-using-herons-formula-let--a--7---b--10---c--13--the-semi-perimeter--s--is-72697862026-04-20T23:48:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-gears-align-we-need-the-least-common-multiple-lcm-of-their-number-of-teeth-24-and-36-first-factor-both-numbers-92952382026-04-20T23:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-least-common-multiple-lcm-of-6-8-and-12-factor-each-number-45321672026-04-20T23:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-geometric-sequence-with-first-term-a--1-and-common-ratio-r--3-the-total-number-infected-by-the-end-of-the-5th-generation-is-the-sum-of-the-first-5-terms-44959822026-04-20T23:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-list-the-positive-factors-of-24-1-2-3-4-6-8-12-24-but-digits-must-be-single-digit-numbers-09-so-valid-digits-are-1-2-3-4-6-8-89520952026-04-20T23:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-the-three-expressions-is-6106592026-04-20T23:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-compute-the-least-common-multiple-lcm-of-7-and-11-then-add-1-since-7-and-11-are-coprime-extlcm7-11--7-imes-11--77-adding-1-gives-77--1--78-the-two-digit-number-is-oxed78-97163112026-04-20T23:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-prime-factorizations-15--3-imes-5-20--22-imes-5-lcm-is-22-imes-3-imes-5--60-the-synchronized-cycle-length-is-oxed60question-a-civil-engineer-designs-a-parabolic-arch-with-the-equation--y---ax2--bx--c--if-the-archs-highest-point-is-at--3-10--and-passes-through--0-4--find-the-value-of--a--94108622026-04-20T23:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-vertex-form-of-a-parabola-is--y---ax---h2--k--where--h-k--is-the-vertex-given-the-vertex--3-10--the-equation-becomes--y---ax---32--10--substituting-the-point--0-4--17082852026-04-20T23:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression--left-mathbfv-cdot-mathbfw-times-mathbfu-right--depends-on-the-magnitude-of-the-binormal-vector-compute--mathbfw-times-mathbfu--leftlangle--1--2-1-rightrangle--so-46797912026-02-23T12:37:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-terms-of-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--where--a--is-the-middle-third-term-and--d--is-the-common-difference-95224102026-04-21T00:24:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--be-the-price-of-one-tool-and--b--be-the-price-of-one-bead-61712612026-04-21T00:24:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-terms-of-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--65487462026-04-21T00:24:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-add-the-two-lengths-and-divide-by-2-first-convert-the-mixed-numbers-to-improper-fractions-69138762026-04-21T00:24:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-recall-that--92794752026-04-21T00:24:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-cost-of-one-unit-of-x-in-dollars-and--y--be-the-cost-of-one-unit-of-y-in-dollars-the-system-of-equations-is-35735282026-04-21T00:32:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-scale-is--frac32-text-cm16-text-km--frac02-text-cm1-text-km--using-this-scale-39605382026-04-21T00:32:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-square-be--s--the-perimeter-of-a-square-is--4s--so-we-have-90944532026-04-21T00:36:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with--b1--1--and-compute-successive-terms-using--eu--u---fracu44--92902302026-04-21T00:36:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-rectangle-have-length--l--and-width--w--the-perimeter-is-42743362026-04-21T00:36:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--sqrtu--so--u--x2--and-the-equation-becomes-4935332026-04-21T00:36:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with--c1--1--and-compute-61171762026-04-21T00:36:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-the-five-values-is-given-by-38695162026-04-21T00:36:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-two-functional-equations-69536472026-04-21T00:45:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-let-the-triangle-have-sides-a--13-b--14-and-c--15-the-semi-perimeter-is-29169082026-04-21T00:45:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-right-triangle-with-legs-a-b-and-hypotenuse-z-the-inradius-is-given-by-48627522026-04-21T00:45:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-hexagon-the-side-length-is-equal-to-the-radius-of-the-circumscribed-circle-thus-the-radius--r--5--cm-the-circumference--c--of-the-circle-is-given-by--c--2pi-r--substituting--c--2pi-times-5--10pi--77632802026-04-21T00:57:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-radius-of-the-orbit-is-the-sum-of-the-planets-radius-and-the-satellites-orbital-radius--6000--10000--16000--km-the-circumference--c--is-calculated-as--c--2pi-times-16000--32000pi--km-96865072026-04-21T00:57:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-total-gb-used-the-cost-equation-is--12-times-20--008x---20--3840--simplify--240--008x---16--3840--combine-terms--2384--008x--3840--subtract-2384--008x---1996--divide-by-008--x---24875--this-negative-value-indicates-an-inconsistency-suggesting-the-business-used-fewer-than-20-gb-recalculate-without-the-008-overage--12x--3840----x--32--verify-12--32--3840-final-answer--boxed32--gb-43695322026-04-21T01:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-maximum-occurs-at-the-vertex--x---fracb2a---frac182-3--3--substitute--x--3--into--y---y---39--183--20---27--54--20--47--the-fertilizer-amount-maximizing-yield-is--boxed3--kg-99134302026-02-23T13:50:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--8---s--frac6sqrt8--frac62sqrt2--frac3sqrt2--rationalize--frac3sqrt22--final-answer--boxeddfrac3sqrt22--46755562026-04-21T01:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-y-intercept-occurs-when--x--0--substitute--t--frac40--10---2--frac1-2---frac12--the-y-intercept-is--boxedleft0--dfrac12right--52560102026-04-21T01:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-minimum-occurs-at--x---fracb2a---frac-2025--2--verify--w2--54---202--45--20---40--45--25--the-smallest--x--is--boxed2-question-a-hydrologist-models-groundwater-flow-with-the-function--fx--sec-x--cos-x2--csc-x---sin-x2--for--0--x--fracpi2--find-the-minimum-value-of--fx--92650812026-04-21T01:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-function-22362642026-04-21T01:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-function--st--left-sinleftfrac3pi7-tright--cosleftfrac4pi7-tright-right--has-period--t--where-both-components-repeat-the-fundamental-period-is-the-least-common-multiple-of-the-periods-of--sinleftfrac3pi7-tright--and--cosleftfrac4pi7-tright--48081562026-04-21T01:13:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the--y--intercept--b--substitute-the-point--10-2--into-the-equation-17532802026-04-21T01:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-maximum-of--px---x2--4x--m--occurs-at-the-vertex-the--x--coordinate-of-the-vertex-is--x--frac-b2a--frac-4-2--2--substitute--x--2--into--px--56052312026-04-21T01:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-equations-using-the-given-points-29946702026-04-21T01:41:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-numerator-and-denominator-by-the-conjugate--sqrt7--sqrt3--9775872026-04-21T01:41:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-de-moivres-theorem--zn--cosleftnthetaright--i-sinleftnthetaright--here--z--cosleftfracpi6right--i-sinleftfracpi6right--so--z6--cosleft6-cdot-fracpi6right--i-sinleft6-cdot-fracpi6right--cospi--i-sinpi--evaluating--cospi---1--and--sinpi--0--so--z6---1--0i--24261352026-04-21T02:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--frac18d0--d0-cdot-2-t12--divide-both-sides-by--d0---frac18--2-t12--note--frac18--2-3--so--2-3--2-t12--equate-exponents---3---fract12--solve-for--t---t--36--34003082026-04-21T02:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-occurs-where-the-vector-from--5-7--to-the-line-is-perpendicular-to-the-direction-vector--beginpmatrix-3--4-endpmatrix--let-the-point-on-the-line-be--2--3t--1--4t--the-vector-from--5-7--to-this-point-is--beginpmatrix-2--3t---5---1--4t---7-endpmatrix--beginpmatrix--3--3t---8--4t-endpmatrix--dot-product-with--beginpmatrix-3--4-endpmatrix--must-be-zero--3-3--3t--4-8--4t--0--simplify---9--9t---32--16t--0-rightarrow-25t---41--0-rightarrow-t--frac4125--substitute-back--x--2--3-cdot-frac4125--frac17325---y---1--4-cdot-frac4125--frac15625--45161352026-02-23T14:50:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero--x-cdot-3--2-cdot--1--0--simplify--3x---2--0-rightarrow-x--frac23--91670472026-04-21T02:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-solving--2cos2theta--1--0-rightarrow-cos2theta---frac12--the-general-solutions-for--2theta--are--2theta--120circ--360circ-k--or--2theta--240circ--360circ-k--where--k-in-mathbbz--dividing-by-2--theta--60circ--180circ-k--or--theta--120circ--180circ-k--for--theta-in-0circ-360circ--the-solutions-are--60circ-120circ-240circ-300circ--94485732026-04-21T02:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite--fx--fracsin2-x--4sin-x--sin-x--frac4sin-x--let--y--sin-x-in-0-1--the-function-becomes--fy--y--frac4y--the-derivative--fy--1---frac4y2--has-critical-point-at--y--2--but--y-leq-1--analyze-endpoints-as--y-to-0---fy-to-infty--at--y--1---f1--1--4--5--the-minimum-is--5--32856692026-04-21T02:07:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--vecoa--mathbfa---vecob--mathbfb--then--overrightarrowoc--mmathbfa--nmathbfb--for--overrightarrowoc-perp-mathbfa---mathbfb--their-dot-product-is-zero--mmathbfa--nmathbfb-cdot-mathbfa---mathbfb--0--expand--mmathbfa2---mmathbfa-cdot-mathbfb--nmathbfb-cdot-mathbfa---nmathbfb2--0--substitute--mathbfa--2---mathbfb--3---mathbfa-cdot-mathbfb--2-cdot-3-cdot-cos-60circ--3---4m---3m--3n---9n--0-rightarrow-m---6n--0--choose--n--1--then--m--6--normalize-if-needed-but-the-relation-is--m--6n--for-simplicity-take--n--1---m--6--12862872026-04-24T12:38:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-measurement-be--u--a-positive-multiple-of-5-such-that--u3--1500--35248792026-04-24T12:38:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of--3u2---5u7--and--4u1--is-given-by-96127312026-04-24T12:38:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfv--langle-v1-v2-v3-rangle-compute-mathbfv-times-mathbfb-79559512026-04-21T02:10:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-magnitude-of-the-cross-product-is-given-by-mathbfu-times-mathbfw--mathbfu-mathbfw-sin-theta-here-theta--60circ-6413182026-04-21T02:10:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-a--b--2025-the-greatest-common-divisor-gcda-b-must-divide-a--b--2025-the-maximum-possible-value-of-gcda-b-occurs-when-a-and-b-are-both-multiples-of-the-largest-divisor-of-2025-since-2025--452--34-cdot-52-its-largest-proper-divisor-is-2025--3--675-thus-gcda-b--675-when-a--675-and-b--1350-41190142026-04-21T02:19:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-cubes-formula-is-leftfracnn12right2-for-n--12-the-sum-is-leftfrac12-cdot-132right2--782--6084-dividing-by-13-6084-div-13--468-with-remainder-0-since-13-cdot-468--6084-43401252026-04-21T02:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-least-common-multiple-lcm-of-14-and-21-is-42-the-smallest-three-digit-multiple-of-42-is-42-times-3--126-74929102026-04-21T02:19:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-prime-factorization-16--24-24--23-cdot-3-lcm-takes-the-highest-powers-24-cdot-3--16-cdot-3--48-36429982026-04-21T02:19:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-prime-factorization-108--22-cdot-33-144--24-cdot-32-gcd-takes-the-lowest-powers-22-cdot-32--4-cdot-9--36-29849782026-04-21T02:19:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-and-arranging-instruments-for-display-first-choose-3-telescopes-from-8-inom83-then-choose-2-microscopes-from-5-inom52-after-selection-arrange-the-5-chosen-instruments-in-a-row-5-ways-14066672026-04-21T02:27:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-5-varieties-from-10-inom105-45058662026-04-21T02:27:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-kiosks-are-arranged-in-a-circle-and-rotations-are-considered-the-same-we-fix-one-session-to-eliminate-rotational-symmetry-first-assign-4-out-of-7-kiosks-to-the-sessions-inom74-cdot-4-would-count-linear-arrangements-but-due-to-circular-symmetry-we-divide-by-4-the-number-of-rotations-53405422026-04-21T02:27:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-constant--k--using-the-given-values-72328942026-04-21T02:27:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression--cos-x--isin-x--corresponds-to--eix--on-the-unit-circle-in-the-complex-plane-therefore--cos-x--isin-x--1--eix--1--the-modulus-is--eix--1--using-the-identity--eix--1--sqrtcos-x--12--sin2-x--we-simplify-11877132026-04-24T12:38:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--sin2-x--cos2-x--1--we-write--fx--frac1sin2-x--4cos2-x--frac11--3cos2-x--let--y--cos2-x--so--0-leq-y-leq-1--then--fx--frac11--3y--as--y--varies-from-0-to-1--1--3y--varies-from-1-to-4-so--fx--varies-from--frac14--to--1--hence-the-range-of--fx--is--leftfrac14-1right--the-final-answer-is--boxedleftfrac14-1right--89368222026-04-22T06:04:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with--tan-z--cot-z--2--recall--tan-z--fracsin-zcos-z---cot-z--fraccos-zsin-z--then-81930572026-04-22T06:04:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-lets-simplify-each-term-first--tan-x--cot-x--fracsin-xcos-x--frac-55588472026-04-21T02:29:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-points-lie-on-the-base-face-of-the-cube-in-the-xy-plane-the-fourth-vertex-of-the-square-face-must-complete-the-rectangle-the-missing-vertex-is-110-as-it-forms-equal-sides-and-right-angles-with-the-given-points-48694652026-04-21T02:34:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-2x--60circ-120circ-420circ-480circ-within-0circ-leq-2x-leq-720circ-thus-x--30circ-60circ-210circ-240circ-sum-30--60--210--240--540circ-84847492026-04-21T02:34:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-scalar-triple-product-equals-the-volume-of-the-parallelepiped-maximum-occurs-when-vectors-are-orthogonal-giving-mathbfa-cdot-mathbfb-times-mathbfc--1-cdot-1-cdot-1-cdot-sin90circcos0circ--1-57547162026-04-21T02:34:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-requires-partitioning-7-distinguishable-solutions-into-3-indistinguishable-non-empty-vials-this-is-equivalent-to-the-stirling-numbers-of-the-second-kind--s7-3--the-formula-for-stirling-numbers-is-16081532026-04-21T02:34:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-modules-are-indistinguishable-and-empty-modules-are-allowed-the-number-of-ways-is-the-sum-of-stirling-numbers-of-the-second-kind-for-partitions-into-1-and-2-non-empty-subsets-69345362026-04-21T02:34:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-ways-to-choose-3-startups--binom63--20--50939622026-04-21T02:34:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-roots-of--fx--x4---6x3--11x2---6x--1--be--r1-r2-r3-r4--we-are-asked-to-find--sumi14-frac1ri--frac1r1--frac1r2--frac1r3--frac1r4--using-the-identity--sum-frac1ri--fracsumijk-ri-rj-rkr-57915192026-04-21T02:38:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-possible-outcomes-for-7-patterns-is-47-to-count-favorable-outcomes-choose-3-patterns-to-show-the-specific-outcome-extreme-heat-in-binom73-ways-and-the-remaining-4-patterns-must-show-one-of-the-other-3-outcomes-giving-34-possibilities-the-probability-is-fracbinom73-cdot-3447--frac35-cdot-8116384--frac283516384-thus-the-final-answer-is-boxeddfrac283516384-79405142026-04-21T02:40:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-possible-state-combinations-is-25--32-the-number-of-ways-to-choose-2-elements-to-be-active-is-binom52--10-since-each-combination-is-equally-likely-the-probability-is-frac1032--frac516-thus-the-final-answer-is-boxeddfrac516-40367962026-04-21T02:40:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-possible-calibration-configurations-is-36-to-count-favorable-cases-choose-4-sensors-to-be-in-precision-mode-in-binom64-ways-and-the-remaining-2-sensors-can-be-in-either-of-the-other-2-settings-giving-22-possibilities-the-probability-is-fracbinom64-cdot-2236--frac15-cdot-4729--frac60729--frac20243-thus-the-final-answer-is-boxeddfrac20243question-a-tech-startup-founder-wants-to-align-two-project-timelines-with-cycles-of-10-and-15-months-what-is-the-least-number-of-months-after-which-both-projects-will-synchronize-85244162026-04-21T02:40:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-projects-synchronize-compute-the-least-common-multiple-lcm-of-10-and-15-factorize-10--2-times-5-and-15--3-times-5-the-lcm-is-2-times-3-times-5--30-thus-the-projects-will-align-after-boxed30-months-2193952026-04-21T02:40:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-day-both-patterns-coincide-is-the-lcm-of-9-and-12-factorize-9--32-and-12--22-times-3-the-lcm-is-22-times-32--36-the-patterns-coincide-on-day-boxed36-60281982026-04-21T02:40:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-the-equation-as-fracx216--fracy24--1-possible-integer-values-for-y-are--2--1-0-1-2-for-each-y-solve-for-x-65854172026-02-23T15:26:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-be-n-then-n-equiv-3-pmod7-and-n-equiv-5-pmod6-solve-the-system-10797352026-04-21T02:40:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-identity-gcd2a---1-2b---1--2gcda-b---1-compute-gcd12-18--6-thus-gcd212---1-218---1--26---1--64---1--boxed63-73404642026-04-21T02:40:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-lcm-of-11-and-13-is-143-the-smallest-three-digit-multiple-of-143-is-143-times-1--143-since-143-times-2--286-is-also-three-digit-but-143-is-smaller-thus-the-answer-is-boxed143question-two-marine-biologists-dr-coral-and-dr-tide-arrive-at-a-research-station-at-random-times-between-700-and-800-if-dr-coral-arrives-after-dr-tide-what-is-the-probability-that-dr-tide-arrived-before-745-6857902026-04-21T02:40:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arrival-times-of-dr-tide-and-dr-coral-be--x--and--y--respectively-where--x-y-in-0-60--minutes-after-700-the-condition--y--x--defines-a-triangular-region-in-the--xy--plane-with-area--frac6022--1800--the-favorable-region-is-where--x--45--and--y--x--integrating-over--x-in-0-45--the-area-is--int045-60---x--dx--60-cdot-45---frac4522--2700---10125--16875--the-conditional-probability-is--frac168751800--frac34--79597332026-04-21T02:40:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-breakthroughs-is--binom203--1140--the-number-of-favorable-outcomes-is--binom52-cdot-binom151--10-cdot-15--150--the-probability-is--frac1501140--frac538--53686322026-04-21T02:40:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-3-flavors-is--binom73--35--and-2-toppings-is--binom42--6--multiplying-gives--35-cdot-6--210--combinations-7705322026-04-21T02:40:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-the-stirling-number-of-the-second-kind--s6-3--calculated-as--frac13-left-36---3-cdot-26--3-cdot-16-right--frac729---192--36--frac5406--90--50342392026-04-21T02:40:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-items--4--6--2--12--total-ways-to-choose-2--binom122--66--favorable-ways--binom62--15--probability--frac1566--frac522--31223022026-04-21T02:40:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-of-the-two-ribbons-we-add-their-lengths-and-divide-by-2-the-total-length-is-32--78--110-inches-dividing-by-2-we-get-frac1102--55-inches-thus-the-average-length-is-boxed55-inches-10714162026-04-21T02:53:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate-the-boundary-values-frac52--25-since-pi-approx-31416-we-have-3pi-approx-94248-we-seek-whole-numbers-n-such-that-25--n--94248-these-are-3-4-5-6-7-8-9-counting-them-there-are-7-whole-numbers-therefore-the-answer-is-boxed7-38141742026-04-21T02:53:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-three-expressions-and-divide-by-3-the-sum-is-3v--4--5v---2--4v--7--3v--5v--4v--4---2--7--12v--9-dividing-by-3-we-get-frac12v--93--4v--3-thus-the-average-is-boxed4v--31---------120-----180-----7800-----30---------61443902026-04-21T02:53:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-starts-at-0-and-increases-by-5-each-time-the-nth-term-is--an--5n-1--setting--5n-1--100--solve-for--n--33631942026-04-21T03:10:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-the-used-amount-from-the-total-12414582026-04-21T03:10:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-the-heights-are-equal-set--hat--hbt--16927782026-04-21T03:39:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify--sqrt081--37121992026-04-21T03:39:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-numerator-and-denominator-by--sqrt3--sqrt2--20696122026-04-21T03:39:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a---units-of-a--b---units-of-b-41901442026-04-21T03:39:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--sin2theta--fracsqrt32--implies--2theta--60circ--360circ-k--or--2theta--120circ--360circ-k--for-integer--k--solving-for--theta--we-get--theta--30circ--180circ-k--or--theta--60circ--180circ-k--within--0circ-360circ--the-solutions-are--30circ-60circ-210circ-240circ--thus-the-answer-is--boxed30circ-boxed60circ-boxed210circ-boxed240circ--71725962026-04-21T03:40:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--mathbfm--beginpmatrix-a--b--c--d-endpmatrix--multiplying--mathbfm--with--beginpmatrix-1--2--3--4-endpmatrix--gives-61083692026-04-21T03:40:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-62351832026-04-21T03:40:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--theta--be-the-angle-between--mathbfa--and--mathbfb--so--costheta--frac12-rightarrow-theta--60circ--let--phi--be-the-angle-between--mathbfb--and--mathbfc--so--cosphi--fracsqrt32-rightarrow-phi--30circ--to-maximize--mathbfa-cdot-mathbfc--cosalpha--where--alpha--is-the-angle-between--mathbfa--and--mathbfc--arrange--mathbfa-mathbfb-mathbfc--in-a-plane-the-maximum-occurs-when--mathbfa--and--mathbfc--are-aligned-but-constrained-by-their-angles-relative-to--mathbfb--the-minimum-angle-between--mathbfa--and--mathbfc--is--60circ---30circ--30circ--so--cos30circ--fracsqrt32--however-if-they-are-aligned--alpha--0circ--but-this-requires--theta--phi--0circ--which-contradicts-the-given-dot-products-instead-use-the-cosine-law-for-angles--cosalpha-leq-cos60circ---30circ--cos30circ--fracsqrt32--thus-the-maximum-is--boxedfracsqrt32--73279402026-04-22T06:06:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--mathbfv--beginpmatrix-a--b--c-endpmatrix--the-cross-product-is-81883942026-04-22T06:06:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-cone-is-given-by--vtextcone--frac13pi-r2-h--substituting--h--2r--we-get-59663592026-04-21T03:52:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-cylinder-is--vtextcyl--pi-r2-h--for-the-cylinder--r--3r--and--h--4r--so-20676822026-04-21T03:53:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-hemisphere-is--vtexthemisphere--frac23-pi-r3--54059432026-04-21T03:53:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-triangle-is-42068512026-04-21T03:53:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-right-triangle-the-circumradius--r--is-half-the-hypotenuse-26101582026-04-21T03:53:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-power-output-we-sum-the-three-expressions-and-divide-by-3-3534032026-04-21T03:53:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-volume-we-add-the-three-fractions-first-find-a-common-denominator-the-least-common-denominator-of-4-8-and-16-is-16-89299072026-04-21T03:53:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-efficiency-we-sum-the-three-expressions-and-divide-by-3-81934302026-04-21T03:53:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve--t2---5t--6--0--factor--t---2t---3--0--the-solutions-are--t--2--and--t--3--thus-the-times-are--oxed2--and--oxed3--87683052026-04-21T04:20:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-vector-mathbfv--eginpmatrix-v1--v2--v3-endpmatrix-such-that-mathbfv-imes-mathbfc--mathbfd-compute-the-cross-product-mathbfv-imes-mathbfc-93457892026-04-21T04:23:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-k-such-that-mathbfp--kmathbfq-is-perpendicular-to-mathbfr-we-use-the-dot-product-condition-23913262026-04-21T04:29:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfw--langle-w1-w2-w3-11129052026-04-21T04:30:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-line-through-a-and-b-is-parametrized-as-95476682026-04-24T12:38:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-equation-is--72204452026-04-21T04:30:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-tetrahedron-all-edges-are-equal-compute-distances-86357932026-04-21T04:30:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides--x2--2--rac1x2--25--so--x2--rac1x2--23--multiply-by-2--2x2--rac2x2--46--oxed46-71173092026-04-21T04:33:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--3x--1---2x--11--solve-for--x---5x--10----x--2--substitute-into--y--32--1--7--the-intersection-is--2-7--oxed2-7-84180422026-04-21T04:33:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--m--n--8--into-the-first-equation--m-cdot-8--24--so--m--3--oxed3question-a-sequence-of-six-real-numbers-forms-an-arithmetic-progression-the-sum-of-the-first-three-terms-is-12-and-the-sum-of-the-last-three-terms-is-30-what-is-the-sum-of-all-six-terms-37418932026-04-21T04:33:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-six-terms-of-the-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d-a--3d--66000852026-04-21T04:33:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-group-the-expression-38188802026-04-21T04:33:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-yields-be--a-ar-ar2-ar3--where--r--0--37991002026-04-21T04:34:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--sqrtu--x--so--u--x2--substituting-into-the-equation-gives-14255242026-04-21T04:34:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--fraca--2ba---2b--then-the-equation-becomes-56417552026-04-21T04:35:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-difference--fx1---fx--is-linear--fx--must-be-a-quadratic-polynomial-let--fx--ax2--bx--c--then-56377262026-04-21T04:35:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-lets-plug-in--a--b--1--into-the-functional-equation-80886702026-04-21T04:35:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--x--y--4--represents-a-diamond-rhombus-centered-at-the-origin-in-the-first-quadrant-it-becomes--x--y--4--with-intercepts-at--4-0--and--0-4--the-full-figure-is-symmetric-across-both-axes-9546502026-04-21T04:35:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-an-equilateral-triangle-is-given-by--fracsqrt34-s2--where--s--is-the-side-length-setting--fracsqrt34-s2--36sqrt3--we-solve-for--s2--144--so--s--12--textcm--the-new-side-length-is--12---4--8--textcm--the-new-area-is--fracsqrt34-times-82--16sqrt3--textcm2--the-decrease-in-area-is--36sqrt3---16sqrt3--20sqrt3--textcm2--boxed20sqrt3-10936842026-04-21T04:35:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-is--frac43pi-y3--and-the-volume-of-a-hemisphere-is--frac23pi-3y3--frac23pi-times-27y3--18pi-y3--the-ratio-is--fracfrac43pi-y318pi-y3--frac454--frac227--boxeddfrac227-31573222026-04-21T04:35:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--frac10--13--152--19--textcm--the-area-is--sqrt1919-1019-13-75531792026-04-21T04:35:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-4-modules-from-12-is-binom124-the-number-of-favorable-outcomes-2-non-compliant-and-2-compliant-is-binom32-cdot-binom92-thus-the-probability-is-fracbinom32-cdot-binom92binom124--frac3-cdot-36495--frac108495--frac1255-boxeddfrac1255-10449172026-04-21T04:57:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-of-selecting-no-quantum-dots-all-nanotubes-is-fracbinom73binom123--frac35220-the-complement-gives-the-probability-of-at-least-one-quantum-dot-1---frac35220--frac185220--frac3744-boxeddfrac3744-46406392026-04-21T04:57:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-select-5-startups-is-binom95-the-favorable-case-is-binom43-cdot-binom52-3-solar-and-2-non-solar-thus-the-probability-is-frac4-cdot-10126--frac40126--frac2063-boxeddfrac2063-question-1-25003672026-04-21T04:57:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-of-full-pieces-divide-the-total-length-by-the-length-of-each-piece-26187042026-04-21T04:58:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-the-four-times-is-given-by-6054892026-04-22T06:07:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-of-the-three-expressions-we-first-add-them-together-84503982026-04-21T05:44:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-we-add-the-two-lengths-and-divide-by-2-92186302026-04-21T05:44:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-44780902026-04-21T05:44:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-determine-all-positive-integers-less-than-or-equal-to-50-that-are-factors-of-50-52564142026-04-21T05:44:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-both-sides-of-the-equation-88872992026-04-21T05:45:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-aim-to-minimize-2301742026-04-21T05:46:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--mathbfa--langle-1-2-3-rangle---mathbfb--langle-1--2-1-rangle--we-are-to-find--mathbfv--such-that-75123622026-02-23T18:58:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-if-the-drone-uses--frac38--of-its-battery-for-45-minutes-then-each--frac18--of-the-battery-powers--frac453--15--minutes-thus-the-remaining-battery-is--frac58--which-powers--5-times-15--75--minutes-therefore-the-remaining-flight-time-is--boxed75--minutes-48442052026-04-21T05:46:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-weekly-growth-by-the-number-of-weeks-70545892026-04-21T05:46:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-the-amount-poured-out-from-the-total-21951752026-04-21T05:46:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula--texttime--fractextdistancetextspeed--30630542026-04-21T05:46:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-normal-growth-by-the-growth-factor-29755372026-04-21T05:46:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-acid---3-times-02--5-times-04--06--2--26--liters-69883692026-04-21T05:46:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-list-is-simply-the-positive-integers-47150182026-04-21T05:46:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--h--be-the-number-of-heads-in-6-flips-of-a-fair-coin-so--h-sim-textbinomial6-frac12--let--d--be-the-outcome-of-a-fair-6-sided-die-uniformly-distributed-over--123456--we-seek--ph--d--86406852026-04-21T05:47:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-binary-strings-of-length-5-using-e-and-j-or-0-and-1-where-no-two-adjacent-characters-are-both-j-this-is-equivalent-to-counting-binary-strings-without-consecutive-1s-89958772026-04-21T05:47:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-coefficient-problem-we-are-arranging-12-fruits-where-67869842026-04-21T05:47:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--v--is-a-positive-multiple-of--5--and--v3--1700--we-want-to-find-the-largest-such--v--47712562026-04-21T05:47:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-requires-partitioning-6-distinguishable-initiatives-into-3-non-empty-indistinct-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s6-3--the-formula-for--sn-k--is--frac1k-sumi0k--1k-i-binomki-in--plugging-in--n6--and--k3--91564832026-04-21T06:01:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-filters-are-identical-and-samples-are-distinguishable-this-is-equivalent-to-partitioning-8-distinct-elements-into-up-to-5-indistinct-subsets-the-total-number-of-ways-is-the-sum-of-stirling-numbers-of-the-second-kind-for--k--1--to--5--31719252026-04-21T06:01:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs--a---b--hypotenuse--c--the-inradius-is--r--fraca--b---c2--using-the-pythagorean-theorem--c--sqrta2--b2--substitute--b--frac2r--c---a1--from-rearranging-the-inradius-formula-squaring-and-solving-yields--c--a--frac2r2--2ar2r--2a--simplifying--c--fraca2r--ar--thus-the-hypotenuse-is--boxeddfracaa--2rr--98882732026-04-21T06:44:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-lateral-surface-area-of-a-cylinder-is--2pi-r-h--the-area-of-a-circle-with-radius--2r--is--pi-2r2--4pi-r2--setting-them-equal--2pi-r-h--4pi-r2--cancel--2pi-r--to-get--h--2r--thus-the-ratio--frachr--boxed2--98809502026-04-21T06:44:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-circles-diameter-equals-the-squares-side-so-radius--r--fracs2--circle-area--pi-leftfracs2right2--fracpi-s24--square-area--s2--ratio--fracfracpi-s24s2--fracpi4--the-ratio-is--boxeddfracpi4--38634382026-04-21T06:44:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-original-side-be--s--original-area--fracsqrt34-s2--new-side--s---3--new-area--fracsqrt34-s---32--the-difference--fracsqrt34-s2---s---32--15sqrt3--simplify--fracsqrt34-6s---9--15sqrt3--cancel--sqrt3--and-solve--frac6s---94--15--leading-to--6s---9--60--so--s--frac696--115--original-side-length-is--boxed115--textcm--13408412026-04-21T06:44:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sphere-surface-area--4pi-r2--circle-area--pi-sqrt2r2--2pi-r2--setting-equal--4pi-r2--2pi-r2--this-implies--4--2--a-contradiction-thus-no-solution-exists-unless-the-circles-radius-is-adjusted-however-if-the-problem-states-equivalence-the-only-possibility-is--r--0--which-is-trivial-rechecking-the-question-reveals-a-misstatement-assuming-the-circles-radius-is--r--then--4pi-r2--pi-r2-rightarrow-r--2r--the-original-questions-setup-is-inconsistent-but-if-forced--r--fracr2--so--boxedr--dfracr2--25367952026-04-21T06:44:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--mt--t---fract33--and--b1--1--compute-successive-terms-70859892026-04-21T06:50:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--sqrtx--so--x--u2--and--px--u2-cdot-u---5u2--6u--u3---5u2--6u--71365412026-04-21T06:50:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--st--t---fract55---c1--frac12--compute-recursively-94248292026-04-21T06:50:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--hx--ax2--bx--c--from--h1--a--b--c--5--and--h-1--a---b--c--3--adding-gives--2a--2c--8--so--a--c--4--the-sum-of-roots-is---racba--4--so--b---4a--substituting--b---4a--into--a--b--c--5---a---4a--c--5-----3a--c--5--since--a--c--4--subtracting-gives---4a--1--so--a---rac14--then--c--4---a--4--rac14--rac174--and--b---4a--1--thus--hx---rac14x2--x--rac174--multiplying-by-4-to-eliminate-fractions--hx---x2--4x--17--verifying--h1---1--4--17--20--wait-inconsistency-rechecking--a---14---c--174---b--1--then--h1---14--1--174---1--4--174--204--5--correct--h-1---14--1--174----1--4--17-4--124--3--correct-sum-of-roots---ba---1---14--4--correct-final-answer--oxed-x2--4x--dfrac174--or--oxed-dfrac14x2--x--dfrac174--22674892026-02-23T20:10:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--x--y--1---k2--k1--k1---2k1-cdot-1--1--1---21--0--verify-consistency-try--x--2-y--1---k3--k2--k1---2k2--0--1---0--1--try--x--y--2---k4--k2--k2---2k4----k4--2k4--0--0----3k4--0----k4--0--assume--kx--0--for-all--x--but--k1--1-66532502026-04-21T06:53:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-note--x2---2x--1--x---12--use-polynomial-division-or-remainder-theorem-for-repeated-roots-let--fx--x5---3x3--2x---1--the-remainder--rx--has-degree--2-so--rx--ax--b--since--x---12--divides--fx---rx--we-have--f1--r1--and--f1--r1--compute--f1--1---3--2---1---1---fx--5x4---9x2--2--so--f1--5---9--2---2---rx--ax--b--so--r1--a--b---1---rx--a--so--a---2--then---2--b---1----b--1--thus-remainder-is---2x--1--final-answer--oxed-2x--1-question-a-plant-biologist-is-studying-a-genetic-trait-that-appears-in-every-12th-plant-in-a-rows-of-crops-planted-in-a-120-plant-grid-if-the-trait-is-expressed-only-when-the-plants-position-number-is-relatively-prime-to-12-how-many-plants-in-the-first-120-positions-exhibit-the-trait-87745422026-02-23T20:10:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-how-many-integers-from-1-to-120-are-relatively-prime-to-12-this-is-given-by-eulers-totient-function-applied-to-the-set-of-positions-but-since-the-condition-is-relative-primality-with-12-we-compute--phi12--the-number-of-integers-from-1-to-12-that-are-coprime-to-12-and-then-multiply-by-the-number-of-full-cycles-in-120-77263352026-04-21T06:53:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-two-gears-will-realign-at-their-starting-position-after-a-time-equal-to-the-least-common-multiple-lcm-of-their-rotation-periods-we-compute--extlcm18-30--57138872026-04-21T06:54:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-with-sides-9-12-and-15-is-a-right-triangle-since-92--122--81--144--225--152-the-area-is-frac12-times-9-times-12--54-the-longest-altitude-corresponds-to-the-shortest-side-9-units-using-the-formula-for-altitude-h--frac2-times-textareatextbase-we-calculate-h--frac2-times-549--12-boxed12-53512642026-04-21T06:57:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-capsule-consists-of-a-cylinder-of-height-10---2-times-2--6-units-subtracting-the-hemispheres-and-two-hemispheres-equivalent-to-one-sphere-volume-of-the-cylinder-pi-r2-h--pi-times-22-times-6--24pi-volume-of-the-sphere-frac43pi-r3--frac43pi-times-8--frac323pi-total-volume-24pi--frac323pi--frac1043pi-boxeddfrac1043pi-61925602026-04-21T06:57:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-herons-formula-the-semi-perimeter-s--frac7--8--92--12-the-area-is-sqrt1212-712-812-9--sqrt12-times-5-times-4-times-3--sqrt720--12sqrt5-the-inradius-r--fractextareas--frac12sqrt512--sqrt5-boxedsqrt5-64369042026-04-21T06:57:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-herons-formula-s--frac10--13--142--185-area--sqrt185185-10185-13185-14--sqrt185-times-85-times-55-times-45-simplify-185-times-45--8325-85-times-55--4675-so-area--sqrt8325-times-4675-approx-sqrt38909375-approx-6238-the-shortest-altitude-corresponds-to-the-longest-side-14-units-h--frac2-times-623814-approx-891-exact-calculation-yields-h--frac2-times-sqrt185-times-85-times-55-times-4514-simplify-the-expression-under-the-square-root-185-times-45--8325-85-times-55--4675-product--38909375-exact-area-frac14-sqrt185--10--13-185--10--13185---10--13185--10---13--frac14-sqrt415-times-45-times-215-times-55-this-is-complex-but-using-exact-values-the-altitude-simplifies-to-frac8414--6-however-precise-calculation-shows-the-exact-area-is-84-so-h--frac2-times-8414--12-wait-conflicting-results-correct-approach-for-sides-10-13-14-semi-perimeter-s--185-area--sqrt185-times-85-times-55-times-45--sqrt38909375-approx-6238-shortest-altitude-is-opposite-the-longest-side-14-h--frac2-times-623814-approx-891-however-exact-form-is-complex-alternatively-using-the-formula-for-altitude-h--frac2-times-textarea14-given-complexity-the-exact-value-is-frac2-times-sqrt3890937514--fracsqrt389093757-but-for-simplicity-assume-the-exact-area-is-84-if-sides-were-13-14-15-but-not-here-given-time-the-correct-answer-is-boxed12-if-area-is-84-altitude-is-12-for-side-14-but-actual-area-is-6238-so-this-is-approximate-for-an-exact-answer-recheck-using-herons-formula-185-times-85-times-55-times-45--frac372-times-frac172-times-frac112-times-frac92--frac37-times-17-times-11-times-916--frac6227116-area--fracsqrt622714-approximate-sqrt62271-approx-24954-area-approx-62385-thus-h-approx-frac1247714-approx-891-the-exact-form-is-fracsqrt6227114-however-the-problem-likely-expects-an-exact-value-so-the-altitude-is-boxeddfracsqrt6227114-or-simplified-further-if-possible-for-practical-purposes-the-answer-is-approximately-891-but-exact-form-is-complex-given-the-discrepancy-the-question-may-need-adjusted-side-lengths-for-a-cleaner-solution-52160082026-02-23T20:13:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--x2--3--then--x2--y---3--substituting-into-the-given-equation-96122272026-04-21T07:07:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-the-equation-by-completing-the-square-group--x--and--y--terms-52537402026-04-21T07:07:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-partial-fractions-to-decompose--frac1kk1k2--let-3575662026-04-21T07:07:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-of-elevation-is--45circ--and-the-adjacent-side-distance-from-the-boat-to-the-base-is-100-meters-using--tan45circ--fractextoppositetextadjacent--we-have--1--frach100--solving-gives--h--100--meters-thus-the-height-of-the-lighthouse-is-boxed100-meters-9207382026-04-21T07:11:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-sphere-is--frac43pi-2x3--frac323pi-x3--the-volume-of-the-hemisphere-is--frac23pi-3x3--18pi-x3--the-ratio-is--frac32318--frac3254--frac1627--the-ratio-of-their-volumes-is-boxeddfrac1627-70786662026-04-21T07:11:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-triangle-is--frac12-times-9-times-12--54--km-the-altitudes-corresponding-to-each-side-are--frac2-times-549--12---frac2-times-5412--9--and--frac2-times-5415--72--the-shortest-altitude-is--72--km-or--frac365--thus-the-length-is-boxeddfrac365-km-81683242026-04-21T07:11:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is--sqrt62--82--10--cm-the-circumference-is--pi-times-textdiameter--10pi--cm-the-circumference-of-the-throughfall-is-boxed10pi-cmquestion-a-bag-contains-7-red-marbles-and-8-blue-marbles-if-four-marbles-are-drawn-at-random-what-is-the-probability-that-exactly-two-are-red-37167932026-04-21T07:11:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-cost-of-an-app-icon-in-dollars-and--y--the-cost-of-a-color-theme-the-system-of-equations-is-53922902026-04-21T07:16:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--p--be-the-price-of-a-pottery-shard-and--s--the-price-of-a-stone-tool-the-system-is-66813712026-04-21T07:16:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--l--be-the-cost-of-a-software-license-and--h--the-cost-of-a-hardware-upgrade-the-system-is-52091752026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by-6-to-eliminate-denominators--4x--4--5x---1--expand--4x--16--5x---5--rearrange--16--5--5x---4x-rightarrow-x--21--the-solution-is--boxed21--40201592026-04-21T07:16:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from--a---b--1--express--a--b--1--substitute-into-the-first-equation--2b--1--3b--14-rightarrow-2b--2--3b--14-rightarrow-5b--12-rightarrow-b--frac125--then--a--frac125--1--frac175--compute--5a--2b--5leftfrac175right--2leftfrac125right--17--frac245--frac1095--the-value-is--boxeddfrac1095-question-a-hydrologist-is-modeling-the-cross-section-of-a-riverbed-approximated-as-an-isosceles-trapezoid-with-bases-of-120-m-and-180-m-and-legs-of-50-m-each-what-is-the-height-of-the-trapezoid-in-meters-72590332026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-trapezoid-have-bases-of-lengths--a--120--m-and--b--180--m-and-equal-legs-of-length--l--50--m-the-height--h--can-be-found-by-dropping-perpendiculars-from-the-ends-of-the-shorter-base-to-the-longer-base-forming-two-right-triangles-on-either-side-and-a-rectangle-in-the-middle-the-difference-in-the-bases-is--b---a--60--m-so-each-of-the-right-triangles-has-a-base-of--frac602--30--m-using-the-pythagorean-theorem-75245642026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-regular-tetrahedron-with-edge-length--e--is-given-by-1922682026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-right-triangle-be--a--15--km-and--b--20--km-the-hypotenuse--c--is-given-by-the-pythagorean-theorem-1129892026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-cones-are-similar-the-ratio-of-their-linear-dimensions-is--frac1560--frac14--the-volume-ratio-of-similar-solids-is-the-cube-of-the-linear-ratio-8904762026-04-21T07:16:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is-12839112026-04-21T07:16:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-sector-of-a-circle-with-radius--r--and-central-angle--theta--in-degrees-is-81542552026-04-21T07:16:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--sqrtx---32--2x---5--simplifies-to--x---3--2x---5--this-gives-two-cases-21519592026-04-21T07:21:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-3-digit-numbers-divisible-by-7-the-smallest-is-105--7-times-15--the-largest-is-994--7-times-142--total-numbers--142---14--128--15538752026-04-21T07:21:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-mackerel-be--m--the-ratio--54--20m--implies--frac54--frac20m--cross-multiplying--5m--80----m--16--there-are--boxed16--mackerel-90581942026-04-21T07:21:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is--frac14--19--233--frac563--simplifying-the-mean-is--boxeddfrac563--question-1-61164652026-04-21T07:21:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-follows-a-binomial-distribution-with-parameters--n--5--and--p--frac13--the-probability-of-exactly-2-successes-is--binom52-leftfrac13right2-leftfrac23right3--calculating-55514812026-04-21T07:29:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-complementary-probability-the-probability-no-sensor-triggers-is--leftfrac34right3--frac2764--subtract-from-1-99627792026-04-21T07:29:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-distinct-6-digit-codes-using-digits-from-1-to-9-with-no-repetition-we-calculate-the-number-of-permutations-of-9-digits-taken-6-at-a-time-54645782026-04-21T07:32:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-there-are-6-work-areas-each-with-2-zones-making-a-total-of-6-times-2--12-zones-since-2-zones-are-selected-at-random-the-total-number-of-ways-to-choose-2-zones-is-77292772026-04-21T07:32:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-code-starts-with-a-letter-so-there-are-26-choices-for-the-first-character-for-the-remaining-4-characters-we-choose-from-the-remaining-25--10--35-characters-since-one-letter-has-already-been-used-with-no-repetition-90742582026-04-21T07:32:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-3-out-of-8-stations-54867252026-04-21T07:32:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x-equiv-1-pmod7-and-x-equiv-1-pmod11-by-the-chinese-remainder-theorem-x-equiv-1-pmod77-the-two-digit-numbers-satisfying-this-are-1--77--78-thus-the-answer-is-boxed78-56922322026-04-21T07:32:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-as-x2--4y2--100-let-x--2k-then-4k2--4y2--100-implies-k2---y2--25-factor-as-k---yk--y--25-the-factor-pairs-of-25-are-125-55--1-25--5-5-solving-3679742026-04-21T07:32:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-recurrence-86519092026-04-21T07:32:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-genetic-markers-in-a-line-where-there-are-3-indistinguishable-markers-from-species-a-3-from-species-b-and-3-from-species-c-since-markers-from-the-same-species-are-indistinguishable-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-404572026-04-21T07:40:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-weather-types-to-7-days-such-that-3-are-rainy-r-2-are-cloudy-c-and-2-are-sunny-s-since-the-days-are-distinct-but-the-types-are-grouped-the-number-of-distinct-sequences-is-the-multinomial-coefficient-54453172026-04-21T07:40:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-6-video-episodes-where-there-are-2-identical-quark-episodes-q-2-identical-lepton-episodes-l-and-2-identical-boson-episodes-b-the-number-of-distinct-release-sequences-is-given-by-85591742026-04-21T07:40:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-note-that-the-sum-of-three-numbers-is-even-if-and-only-if-all-three-are-even-or-exactly-one-is-even-however-the-only-even-prime-is-2-so-for-the-sum-to-be-even-exactly-one-of-the-selected-primes-must-be-2-and-the-other-two-must-be-odd-primes-93807012026-04-21T07:40:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--x--rac1x--3--to-find--x4--rac1x4--we-proceed-step-by-step-23394902026-04-21T07:41:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate--g2--5991052026-04-21T07:41:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-slope--m--of-the-line-45199432026-04-21T07:41:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--x2---y2--2025--factors-as--x---yx--y--2025--since--x--and--y--are-integers-both--x---y--and--x--y--must-be-integers-let--a--x---y--and--b--x--y--so--ab--2025--then--x--raca--b2--and--y--racb---a2--for--x--and--y--to-be-integers--a--b--and--b---a--must-both-be-even-meaning--a--and--b--must-have-the-same-parity-since--2025--34-cdot-52--it-has--4121--15--positive-divisors-each-pair--a-b--such-that--ab--2025--gives-a-solution-but-only-those-with--a--and--b--of-the-same-parity-are-valid-since-2025-is-odd-all-its-divisors-are-odd-so--a--and--b--are-both-odd-ensuring--x--and--y--are-integers-each-positive-divisor-pair--a-b--with--a-leq-b--gives-a-unique-solution-and-since-2025-is-a-perfect-square-there-is-one-square-root-pair-there-are-15-positive-divisors-so-15-such-factorizations-but-only-those-with--a-leq-b--are-distinct-under-sign-and-order-considering-both-positive-and-negative-factor-pairs-each-valid--ab--with--a-76958152026-02-23T20:58:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-events-be-20615452026-04-21T07:46:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-the-gears-align-again-we-compute-the-least-common-multiple-lcm-of-their-rotation-periods-since-they-rotate-at-48-and-72-rpm-rotations-per-minute-the-time-until-alignment-is-the-time-it-takes-for-each-to-complete-a-whole-number-of-rotations-such-that-both-return-to-start-simultaneously-this-is-equivalent-to-the-lcm-of-the-number-of-rotations-per-minute-in-terms-of-cycle-time-first-find-the-lcm-of-the-rotation-counts-over-time-or-convert-to-cycle-periods-the-time-for-one-rotation-is--frac148--minutes-and--frac172--minutes-so-we-find--mathrmlcmleftfrac148-frac172right--frac1mathrmgcd48-72--compute--mathrmgcd48-72--34113022026-04-21T07:50:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-entire-spherical-tumor-modeled-as-a-circle-in-the-cross-section-with-radius-6-cm-is-58132142026-04-21T07:51:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-is-77034422026-04-21T07:51:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-hemisphere-is-92292312026-04-21T07:51:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-area-of-the-region-enclosed-by-the-circle-x2--y2--16-and-the-line-y--x--2-we-first-determine-the-points-of-intersection-substitute-y--x--2-into-the-circle-equation-44754212026-04-21T08:20:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--fx--ax3--bx2--cx--d--using-the-given-values-97630072026-04-21T08:36:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-minimize-the-expression-85572062026-04-21T08:48:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-recurrence-31723072026-04-21T08:48:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--y--intercept--b--represents-the-initial-value-when--x--0--here-the-folding-rate-starts-at-23-units-so--b--23--59263232026-04-21T08:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--s--be-the-sensor-price-and--m--the-meter-price-the-system-is-54632892026-04-21T08:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set---3x--18--4x---2--solve--18--2--7x-rightarrow-x--rac207--substitute-into--y---3-cdot-rac207--18---rac607--rac1267--rac667--intersection-at--left-rac207-rac667-46500772026-04-21T08:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--v--sqrtu--so--u--v2--substituting-into-the-equation-gives--v3---4v2--3v--0--factoring-yields--vv2---4v--3--0--which-simplifies-to--vv-1v-3--0--thus--v--0-1-3--corresponding-to--u--0-1-9--the-sum-of-the-roots-is--0--1--9--10--oxed10-27226972026-04-21T08:52:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--x2---3--so--x2--t--3--substitute-into-the-equation--ft--2t--32---12t--3--13--expanding--2t2--6t--9---12t---36--13--2t2--12t--18---12t---36--13--2t2---5--thus--fx2--1--2x2--12---5--2x4--4x2--2---5--2x4--4x2---3--oxed2x4--4x2---3-7260472026-04-21T08:52:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-sides-be--a--and--b--by-the-pythagorean-theorem--a2--b2--252--625--the-area--a--ab--to-maximize--a--use-the-identity--a--b2--a2--2ab--b2--but-it-is-more-direct-to-note-that-for-fixed--a2--b2---ab--is-maximized-when--a--b--thus--2a2--625-rightarrow-a--rac25sqrt2--the-maximum-area-is--a--leftrac25sqrt2-17545652026-04-21T08:52:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-possible-value-of-mathbfu-cdot-mathbfw-we-start-by-using-the-cauchy-schwarz-inequality-mathbfu-cdot-mathbfw-leq-mathbfu-mathbfw--8-this-gives-the-bound---8-leq-mathbfu-cdot-mathbfw-leq-8-however-we-must-incorporate-the-given-dot-products-mathbfu-cdot-mathbfv--1-and-mathbfv-cdot-mathbfw--6-2438572026-02-23T22:17:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--pt--at2--bt--c--using-the-given-data-93623942026-04-22T06:15:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--et--at3--bt2--ct--d--use-the-given-values-81615442026-04-22T06:15:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-system-81984522026-04-21T09:29:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-sum-63538182026-04-21T09:46:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-a--b--2024-and-suppose-d--gcdab-since-d-mid-a-and-d-mid-b-then-d-mid-a--b--2024-so-d-is-a-divisor-of-2024-4074372026-04-21T09:46:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-sum-of-cubes-32940462026-04-21T09:46:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-property-31233972026-04-21T09:46:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-travel-time-we-compute-the-arithmetic-mean-of-the-two-times-125-minutes-and-8-minutes-the-formula-for-the-average-is-75838082026-04-21T10:20:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-convert-2-hours-into-seconds-87406622026-04-21T10:20:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-spacecraft-travels-from-the-sun-to-venus-67-million-miles-then-from-venus-to-earth-the-remaining-distance-since-earth-is-93-million-miles-from-the-sun-and-venus-is-67-million-miles-away-the-distance-from-venus-to-earth-is-79031932026-04-21T10:20:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-time-until-alignment-is-the-least-common-multiple-lcm-of-88-and-4333-first-factor-both-numbers-17655772026-04-21T10:20:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-12-15-and--x--is-given-by--frac12--15--x3--14--multiply-both-sides-by-3--27--x--42--subtract-27--x--15--boxed15-55051562026-04-21T10:20:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-proportion--frac74--frac21x--cross-multiply--7x--84--divide--x--12--boxed12-98964002026-04-21T10:20:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-integers-between-447-and-707-are-5-6-7-total-3-boxed3-1491062026-04-21T10:20:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-total-days-be--x--ratio--frac38--frac15x--cross-multiply--3x--120--divide--x--40--boxed40-69707902026-04-21T10:20:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-total-articles-be--x---06x--45--divide--x--frac4506--75--boxed75question-if-fraca--ba---b--fraca---ba--b--3-find-the-value-of-fraca2--b2a2---b2-93864432026-04-21T10:20:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-3-out-of-5-distinct-data-types-and-arranging-them-in-order-since-order-matters-and-no-repetition-is-allowed-this-is-a-permutation-problem-the-number-of-permutations-of-5-distinct-items-taken-3-at-a-time-is-given-by-42298602026-04-21T11:30:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-counting-the-number-of-6-day-sequences-using-4-experiment-types-p-c-b-a-where-each-type-appears-at-least-once-and-sequences-are-ordered-this-is-a-classic-inclusion-exclusion-problem-counting-the-number-of-surjective-functions-from-a-6-day-sequence-to-4-experiment-types-74863432026-04-24T12:40:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-code-has-5-positions------the-middle-3rd-character-is-fixed-as-b-we-must-choose-4-distinct-letters-from-a-b-c-d-e-with-b-fixed-in-position-3-so-the-remaining-4-positions-use-the-other-4-letters-but-b-is-already-placed-so-we-use-a-c-d-e-for-the-other-4-positions-75296082026-04-21T11:31:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-tokens-in-total-where-there-are-4-identical-gold-3-identical-silver-and-2-identical-bronze-tokens-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-42132482026-04-21T11:31:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-7-badges-where-3-are-identical-robotics-r-2-are-identical-coding-c-and-2-are-identical-data-science-d-the-number-of-distinct-sequences-is-the-multinomial-coefficient-60152392026-04-21T11:31:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-equation-in-spherical-coordinates-is--81696312026-04-21T11:44:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-vector-equation-mathbfv-imes-mathbfa--mathbfb-has-a-solution-only-if-mathbfa-cdot-mathbfb--0-since-the-cross-product-is-perpendicular-to-mathbfa-42842572026-04-21T11:44:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-identity-sin2z--2sin-z-cos-z-so-85044482026-04-21T11:44:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--3t--9t-1--since--9--32--rewrite-the-equation-as--3t--32t-1--simplify-the-right-hand-side--3t--32t-1--equate-the-exponents--t--2t---1--solve--t--2t---2--so--t--2--verify-by-substituting-back--32--91----9--9--thus-the-time-is--boxed2--96199842026-04-21T11:49:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-liters-of-15-solution-and--y--be-liters-of-5-solution-the-system-is-55012452026-04-21T11:49:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--20--80e-01t--40--subtract-20--80e-01t--20--divide-by-80--e-01t--025--take-natural-logarithm---01t--ln025--solve--t---fracln02501--fracln401-approx-frac138601--1386--round-to-two-decimal-places--boxed1386--minutesthe-analytical-engine-designed-by-charles-babbage-is-considered-a-precursor-to-modern-computers-if-babbages-design-allows-a-program-to-execute-120-distinct-operations-and-each-operation-requires-3-memory-accesses-how-many-total-memory-accesses-are-needed-to-complete-the-program-once-67283612026-04-21T11:49:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-group-and-complete-the-square-75869802026-04-21T11:50:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use--z2--w2--z--w2---2extrez-overlinew--compute--z--w2--2--4i2--4--16--20--let--z-overlinew--a--bi--then--extrez-overlinew--a--from--z--w--2--4i--and--zw--13---2i--note--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--2--4i2---2a--20---2a--also--zw--overlinezw--2extrezw--26--but-this-path-is-complex-alternatively-solve-for--z2--w2--z--w2---2extrez-overlinew--however-using--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--z--w2---2extrez-overlinew--since--z-overlinew--overlinez-w--2extrez-overlinew--and--z--woverlinez--overlinew--z2--w2--z-overlinew--overlinez-w--z2--w2--2extrez-overlinew--let--s--z2--w2--then--20--s--2extrez-overlinew--from--zw--13---2i--take-modulus-squared--zw2--169--4--173--z2-w2--let--z2--a---w2--b--then--a--b--s---ab--173--also--s--20---2extrez-overlinew--this-system-is-complex-instead-assume--z--and--w--are-roots-of--x2---2--4ix--13---2i--0--compute-discriminant--d--2--4i2---413---2i--4--16i---16---52--8i---64--24i--this-is-messy-alternatively-use--z2--w2--z--w2--z---w2---2z-overlinew--but-no-correct-approach--z2--w2--z--woverlinez--overlinew---2extrez-overlinew--20---2extrez-overlinew--from--z--w--2--4i---zw--13---2i--compute--z-overlinew--overlinez-w--2extrez-overlinew--but--z--woverlinez--overlinew--20--z2--w2--z-overlinew--overlinez-w--s--2extrez-overlinew--let--s--z2--w2---t--extrez-overlinew--then--s--2t--20--also--z-overlinew--zw--from--zw--sqrt173--but--t--extrez-overlinew--however-without-more-info-this-is-incomplete-re-evaluate-use--z2--w2--z--w2---2extrez-overlinew--and--extrez-overlinew--extreraczww-overlinew-cdot-overlinew2--too-complex-instead-assume--z--and--w--are-conjugates-but--z--w--2--4i--implies--z--a--bi---w--a---bi--then--2a--2-rightarrow-a--1---2b--4i-rightarrow-b--2--but--zw--a2--b2--1--4--5-69256182026-02-24T01:32:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-rationalize-the-denominator-multiply-numerator-and-denominator-by-the-conjugate-sqrt7--sqrt2-67866172026-04-21T12:04:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with-the-identity-82474432026-04-21T12:04:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-distributive-property-82363172026-04-21T12:04:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--ht--t2---4t--5m--and--h3--10--substitute--t--3--into-the-expression-64410932026-04-21T12:04:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--py--y2---6y--9m--and--p5--22--substitute--y--5--23298422026-04-21T12:04:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--cn--n2---3n--2m--and--c4--14--substitute--n--4--91645702026-04-21T12:05:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--vt--t2---4t--mt--and--v2--8--substitute--t--2--55134672026-04-21T12:05:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product--x-cdot-3--2-cdot--x--3x---2x--x--set--x--0--thus--oxed0--is-the-solution-78552262026-02-24T01:45:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--sin2x--the-equation-becomes--y2--3y--2--0--which-factors-as--y--1y--2--0--thus--y---1--or--y---2--since--sin2x--ranges-between--1-and-1--y---2--is-invalid-for--y---1---sin2x---1--has-infinitely-many-solutions-eg--2x--rac3pi2--2pi-k---k-in-mathbbz--however-if-restricted-to-a-specific-interval-not-stated-here-the-count-would-depend-on-the-domain-assuming--x-in-mathbbr--there-are-infinitely-many-solutions-but-if-the-question-implies-a-general-count-the-answer-is-oxedextinfinitely-many-66978592026-02-24T01:46:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-square-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal--d--of-a-square-with-side-length--8--is--d--8sqrt2--thus-the-radius--r--of-the-circle-is-half-the-diagonal-89029042026-04-21T12:12:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-with-radius--2r--is-59813892026-04-21T12:12:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--s--raca--ba---b--raca---ba--b--combine-the-fractions-87024072026-04-21T12:17:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--px--ax2--bx--c--using-the-given-values-98892942026-04-21T12:17:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-recurrence--an1--an---racan36--resembles-the-taylor-series-for--rctanu--where--racddu-rctanu--rac11--u2--however-the-recurrence-is-not-exact-assume-the-limit--l--exists-then--l--l---racl36-rightarrow-racl36--0-rightarrow-l--0--to-confirm-convergence-note--a1--pi2-pprox-157--1--and--an1--an1---racan26--since--a1--sqrt6---an--is-decreasing-and-bounded-below-by-0-by-monotone-convergence--an-o-0--90106222026-04-21T12:17:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-of-the-circular-lake-be--r--km-the-chord-length-is-10-km-so-half-the-chord-length-is--5--km-the-perpendicular-distance-from-the-center-to-the-chord-is-3-km-by-the-pythagorean-theorem-applied-to-the-right-triangle-formed-by-the-radius-half-the-chord-and-the-perpendicular-from-the-center-to-the-chord-we-have-94620092026-04-21T12:38:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-central-angle-corresponding-to-the-arc-is--120circ--or--rac2pi3--radians-the-chord-length--c--subtended-by-a-central-angle--heta--in-a-circle-of-radius--r--is-given-by-4616932026-04-21T12:38:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-an-equilateral-triangle-with-side--s--the-circumradius--r--is-given-by-76835752026-04-21T12:38:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--r--be-the-radius-of-the-forest-the-chord-length-is-14-km-so-half-is-7-km-the-perpendicular-distance-from-the-center-to-the-chord-is-5-km-using-the-pythagorean-theorem-52153492026-04-22T06:19:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-chord-length--c--1000--km-radius--r--500sqrt2--70231632026-04-22T06:19:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-surface-area-of-a-regular-hexagonal-prism-consists-of-the-area-of-the-two-hexagonal-bases-and-the-six-triangular-lateral-faces-each-face-is-equilateral-with-side-length--s--4--cm-80065592026-04-21T12:52:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter--s--frac10--13--152--19--km-the-area--a--sqrt1919-1019-1319-15--sqrt19-cdot-9-cdot-6-cdot-4--sqrt4104--6407--km-approximate-the-shortest-altitude-corresponds-to-the-longest-side-15-km-using--a--frac12-cdot-textbase-cdot-textheight--the-altitude--h--frac2a15--frac2-cdot-640715-approx-854--km-for-exact-value-simplify--sqrt4104--sqrt4-cdot-1026--2sqrt1026--but-numerical-approximation-suffices-here-thus-the-shortest-altitude-is--boxed854--km-32052792026-04-21T13:00:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-hemisphere-is--frac23pi-r3--frac23pi-2x3--frac163pi-x3--the-cylinders-volume-is--pi-r2-h--pi-x2-cdot-4x--4pi-x3--the-ratio-is--fracfrac163pi-x34pi-x3--frac163-div-4--frac43--thus-the-ratio-is--boxeddfrac43--71653162026-04-21T13:00:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-circles-diameter-using-the-pythagorean-theorem--textdiagonal--sqrt32--42--5--cm-the-circumference-is--pi-cdot-textdiameter--5pi--cm-thus-the-circumference-is--boxed5pi--cm-29977352026-04-21T13:00:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diameter-of-the-semicircular-drum-is-12-meters-so-the-radius-is-half-the-diameter-thus--r--frac122--6--meters-the-radius-is--boxed6--metersquestion-find--x--such-that-the-vectors--beginpmatrix-x--3-endpmatrix--and--beginpmatrix-2---x-endpmatrix--are-orthogonal-16397982026-02-24T02:42:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-equals-zero-compute-the-dot-product--x-cdot-2--3-cdot--x--2x---3x---x--set-this-equal-to-zero---x--0--solving-gives--x--0--boxed0-18618622026-02-24T02:42:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--frac2sin-x--1sin-x---3--solve-for--sin-x---ysin-x---3--2sin-x--1-rightarrow-ysin-x---3y--2sin-x--1-rightarrow-sin-xy---2--3y--1--thus--sin-x--frac3y--1y---2--since--sin-x-leq-1--solve--left-frac3y--1y---2-right-leq-1--this-leads-to---1-leq-frac3y--1y---2-leq-1--sol-27853442026-02-24T02:42:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circumscribed-circle-using-the-pythagorean-theorem-the-diagonal--d--sqrt102--242--sqrt100--576--sqrt676--26--km-the-circumference--c--pi-times-d--26pi--km-thus-the-circumference-is--boxed26pi--75892542026-04-21T13:04:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-incircle-is--pi-c2--for-a-right-triangle-the-semiperimeter--s--fraca--b--z2--and-the-inradius--c--fraca--b---z2--the-area-of-the-triangle-is--r-times-s--c-times-z--c--therefore-the-ratio-is--fracpi-c2cz--c--fracpi-cz--c--the-answer-is--boxeddfracpi-cz--c--47153412026-04-21T13:04:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs-7-and-24-and-hypotenuse-25-the-hypotenuse-is-the-diameter-of-the-circumscribed-circle-the-radius--r--frac252--125--units-thus-the-radius-is--boxed125--36137982026-04-21T13:04:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-be--3x--and-width--2x--the-perimeter--23x--2x--10x--7686702026-04-21T13:04:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-5-distinct-tracking-devices-to-8-bird-species-such-that-each-species-gets-at-least-one-device-this-is-equivalent-to-counting-the-number-of-onto-functions-from-a-set-of-8-species-to-a-set-of-5-devices-with-the-condition-that-no-device-is-left-unused-72080772026-04-22T06:19:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-sequences-of-length-5-one-per-layer-where-each-element-is-an-epoch-from-1-to-4-and-no-two-consecutive-layers-have-the-same-epoch-68347212026-04-21T13:06:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-total-number-of-6-digit-numbers-with-digits-only-3-or-7-each-digit-has-2-choices-so--26--64--but-6-digit-numbers-cannot-start-with-0-but-since-digits-are-only-3-or-7-all-64-are-valid-74528142026-04-21T13:06:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply--frac34--by--frac52--52066062026-04-21T13:22:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-original-height-by--frac32--16421502026-04-21T13:22:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract--frac23--from--frac73--59245892026-04-21T13:22:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-simplifies-to--a---3--5--so-22740232026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-4-digit-number-divisible-by-11-is--1001--since--1001-div-11--91--and-the-largest-is--9999--since--9999-div-11--909--the-count-is--909---91--1--819--60614022026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-each-of-the-6-rods-there-are-3-choices-of-connectors-the-total-configurations-are-36--729-therefore-the-answer-is-boxed729-16601582026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-items-is-binom163--560-the-favorable-cases-are-binom72-times-binom91--21-times-9--189-the-probability-is-frac189560--frac2780-thus-the-answer-is-boxeddfrac2780-87717072026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-ways-to-select-4-items-binom254--12650-the-unfavorable-case-no-turtles-is-binom154--1365-the-probability-of-at-least-one-turtle-is-1---frac136512650--1---frac2732530--frac22572530-the-answer-is-boxeddfrac22572530-78623962026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-items-are-5--7--12-with-5-red-and-7-blue-the-number-of-distinct-sequences-is-frac1257--792-therefore-the-answer-is-boxed792question-a-robotic-engineer-is-programming-a-machine-to-assemble-components-using-exactly-3-different-colored-10440262026-04-21T13:22:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-6-identical-panels-into-4-distinguishable-rooftops-with-each-rooftop-receiving-at-least-one-panel-this-is-a-classic-stars-and-bars-problem-with-the-constraint-of-positive-integers-5007742026-04-21T13:51:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-organisms-where-there-are-repeated-types-4-thermophiles-t-3-psychrophiles-p-and-2-barophiles-b-the-number-of-distinct-permutations-of-a-multiset-is-given-by-42338172026-04-21T13:51:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-identify-the-prime-numbers-on-a-six-sided-die-2-3-5--3-primes-non-primes-1-4-6--3-non-primes-48867392026-04-21T13:51:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve--2sin2z--1--0-rightarrow-sin2z---rac12--the-general-solutions-for--2z--are--2z--210circ--360circ-k--and--2z--330circ--360circ-k--where--k-in-mathbbz--dividing-by-2--z--105circ-165circ-285circ-345circ--within--0circ-360circ--the-sum-is--105--165--285--345--900circ--oxed900-41782832026-04-21T14:03:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-transformation-swaps-components-and-negates-one-the-standard-matrix-is--eginpmatrix-0---1--1--0-endpmatrix--oxedeginpmatrix-0---1--1--0-endpmatrix-88285562026-02-24T03:51:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-minimizes-distance-parametrize-the-line-as--t-2t--1--the-squared-distance-to--3-4--is--t---32--2t--1---42--t---32--2t---32--differentiate--2t---3--8t---12--0-rightarrow-10t---18--0-rightarrow-t--18--substituting-the-point-is--18-46--oxed18-46-55835522026-04-21T14:03:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify--fn--by-factoring-the-numerator--n3---8--n---2n2--2n--4--thus--fn--racn---2n2--2n--4n---2--n2--2n--4--for--n-96972342026-04-21T14:03:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--rac3t4---t2--for--t--2--denominator--4---t2--0--so--y--0--rewrite--y--rac3t-t2---4---rac3tt---2t--2--let--t--2--epsilon---epsilon--0--but-instead-analyze--y--as--t-o-2---y-o--infty--as--t-o-infty---y-o-0---the-minimum-value-of-59422162026-02-24T03:51:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-1-5-9-13-17-21-25-29-33-37-the-sum-of-an-arithmetic-sequence-is-given-by--s--fracn22a--n-1d--here--n--10---a--1---d--4--70875142026-04-21T14:04:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-integers--n--such-that--n-equiv-3-pmod7--among-the-first-200-positive-integers-these-numbers-form-an-arithmetic-sequence--3-10-17-ldots--with-first-term-3-and-common-difference-7-71005052026-04-21T14:04:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-requires-counting-the-number-of-ways-to-partition-4-distinct-items-into-2-non-empty-identical-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s4-2--the-formula-for--sn-k--is--sn-k--sn-1-k-1--k-cdot-sn-1-k--using-known-values--s4-2--7--thus-the-number-of-ways-is--boxed7--23724402026-04-21T14:04:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-guidelines-from-5-is--binom53--10--the-number-of-favorable-outcomes-2-welfare-1-non-welfare-is--binom32-cdot-binom21--3-cdot-2--6--the-probability-is--frac610--frac35--thus-the-probability-is--boxeddfrac35--81051312026-04-21T14:04:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-circumference-of-the-circle-in-which-a-rectangle-is-inscribed-we-first-recognize-that-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-of-a-5-cm-by-8-cm-rectangle-is-81166862026-04-21T14:04:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-standard-trigonometric-values-69447522026-04-21T14:04:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-atextcircle-of-the-inscribed-circle-is-51154122026-04-21T14:04:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-ramanujans-approximation-for-the-circumference-of-an-ellipse-70067282026-04-21T14:04:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-radius-r-of-the-inscribed-circle-of-a-triangle-with-sides-a--13-b--14-and-c--15-we-use-the-formula-13006072026-04-21T14:04:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-define--fu--such-that--fx2--2--x4--4x2--4--observe-that-the-right-hand-side-can-be-rewritten-as-37389472026-04-21T14:08:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--p--q--5--and--pq--6--we-use-the-identity-97178832026-04-21T14:08:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-observe-that-the-right-hand-side-is--x3--12--2--however-we-can-directly-write-89983622026-04-21T14:08:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-compute--x2--x--13-mod-x2---x--1--94324722026-04-21T14:09:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-identify-the-prime-numbers-between-1-and-6-2-3-5-so-there-are-3-primes-and-3-non-primes-1-4-6-on-each-die-27957262026-04-21T14:12:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-partitioning-5-distinguishable-objects-proposals-into-3-indistinguishable-non-empty-subsets-panels-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted-s53-93866892026-04-21T14:12:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-4-distinguishable-satellites-into-2-indistinguishable-orbital-slots-where-each-slot-can-hold-any-number-including-zero-and-the-order-within-a-slot-does-not-matter-since-slots-are-indistinct-and-not-ordered-23078232026-04-21T14:13:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-partitioning-6-distinguishable-objects-proposals-into-3-non-empty-indistinct-subsets-teams-this-is-given-by-the-stirling-number-of-the-second-kind-s63-85438612026-04-21T14:13:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-rewrite-the-equation-in-standard-form-27502282026-04-21T14:20:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is--a--frac3sqrt32s2--given--54sqrt3--frac3sqrt32s2--solving-for--s2--yields--s2--36--so--s--6--the-new-side-length-is--6---2--4--the-new-area-is--frac3sqrt3242--24sqrt3--the-decrease-in-area-is--54sqrt3---24sqrt3--30sqrt3--boxed30sqrt3-71953662026-04-21T14:21:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-circular-sector-is--a--fractheta360-pi-r2--original-area--frac90360-pi-122--36pi--new-radius--12--4--16--meters-new-area--frac90360-pi-162--64pi--the-increase-is--64pi---36pi--28pi--boxed28pi-7862692026-04-21T14:20:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-area--frac12-times-15-times-20--150--km-new-shorter-leg--15--5--20--km-new-area--frac12-times-20-times-20--200--km-the-increase-is--200---150--50--km-boxed50-72995242026-04-21T14:21:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-area--8-times-15--120--m-new-dimensions--8---3--5--meters-and--15---3--12--meters-new-area--5-times-12--60--m-the-decrease-is--120---60--60--m-boxed60question-a-geologist-is-studying-the-periodic-patterns-of-seismic-waves-and-notes-that-a-specific-wave-pattern-repeats-every-24-seconds-if-the-current-time-is-2025-06-28-182116-what-is-the-smallest-positive-time-in-seconds-after-182116-when-the-wave-pattern-will-repeat-such-that-this-time-is-congruent-to-6-mod-11-47674222026-04-21T14:21:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--be-the-time-in-seconds-after-182116-when-the-pattern-repeats-and-satisfies--t-equiv-6-pmod11--since-the-wave-pattern-repeats-every-24-seconds-we-are-looking-for-the-smallest--t--such-that-75512452026-04-21T14:21:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--px--ax3--bx2--cx--d--we-are-given-50967712026-04-21T14:46:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-functional-equation-19089472026-04-21T14:46:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-fourth-vertex-of-a-regular-tetrahedron-with-the-given-three-vertices-a--1-2-3-b--4-5-6-and-c--7-8-9-we-first-compute-the-pairwise-distances-between-the-given-points-78352012026-04-21T14:53:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfv--beginpmatrix-a--b-endpmatrix-be-any-vector-orthogonal-to-beginpmatrix-3---4-endpmatrix-ie-4729102026-04-21T14:53:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with--b1--2--compute--b2--q2--22---rac244--4---rac164--4---4--0--then-compute--b3--q0--02---rac044--0---0--0--thus--b3--oxed0--599062026-04-21T14:55:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve--3k---rac32k2---1--let--x--3k--note--x--0--the-inequality-becomes--x---racx22---1--or---racx22--x--1--0--multiply-by--2-reversing-inequality--x2---2x---2--0--solve--x2---2x---2--0--roots-are--x--1-pm-sqrt3--since--x--0--critical-point-is--x--1--sqrt3-pprox-2732--the-quadratic-is-positive-when--x--1--sqrt3--since--x--3k--find-smallest--k--such-that--3k--1--sqrt3--test--k--1---31--3--2732--check--r1--3---rac92---15---1--thus--k--oxed1--64363652026-02-24T04:43:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-distinct-letters-is-inom263-the-favorable-cases-involve-selecting-3-letters-that-correspond-to-exactly-one-placeholder-symbol-assuming-each-symbol-is-defined-by-a-unique-combination-of-3-letters-with-no-overlap-in-symbol-definitions-there-are-10-favorable-combinations-thus-the-probability-is-rac10inom263--rac102600--rac1260-oxeddfrac1260-42100452026-04-21T14:55:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-two-letters-as-a-single-entity-this-gives-4-arrangements-for-the-entities-the-letter-pair-and-3-manuscripts-the-letters-within-the-pair-can-be-ordered-in-2-ways-total-favorable-arrangements-4-imes-2-total-possible-arrangements-5-the-probability-is-rac4-imes-25--rac25-oxeddfrac25-97284662026-04-21T14:55:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-there-are-11-integers-010-of-which-6-are-even-0-2-4-6-8-10-the-number-of-ways-to-choose-4-distinct-even-integers-is-inom64-the-total-number-of-ways-to-choose-4-distinct-integers-is-inom114-the-probability-is-racinom64inom114--rac15330--rac122-oxeddfrac122-85196212026-04-21T14:55:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-letters-26-vowels-5-consonants-21-total-3-letter-arrangements-with-distinct-letters-26-imes-25-imes-24-unfavorable-cases-no-vowels-21-imes-20-imes-19-favorable-probability-1---rac21-imes-20-imes-1926-imes-25-imes-24--1---rac798015600--1---rac133260--rac127260-oxeddfrac127260-51739882026-04-21T14:55:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-3-18th-century-manuscripts-is-inom53-total-ways-to-choose-3-manuscripts-from-15-is-inom153-the-probability-is-racinom53inom153--rac10455--rac291-oxeddfrac291question-if-the-combined-population-of-two-marine-mammal-species-in-a-region-is-10-and-the-sum-of-their-squared-populations-is-50-find-the-sum-of-their-cubes-25460102026-04-21T14:55:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-populations-be--a--and--b--given--a--b--10--and--a2--b2--50--first-compute--ab--using--a--b2--a2--2ab--b2--substituting--100--50--2ab--so--ab--25--the-sum-of-cubes-is--a3--b3--a--b3---3aba--b--1000---3-cdot-25-cdot-10--1000---750--oxed250--30042412026-04-21T14:55:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--x--y--45--multiplying-by-5-yields--5x--5y--5x--y--5-cdot-45--oxed225--27314662026-04-21T14:55:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-numerator-and-denominator-by--sqrt7--sqrt3--12258982026-04-21T14:55:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-recognize-it-as-a-difference-of-cubes--2a3---3b3--8a3---27b3--alternatively-expand-directly-54532522026-04-21T14:55:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--3x--2---2x--12--solve--5x--10--so--x--2--substitute-into--y--32--2--8--the-intersection-is--oxed2-8-question-let--gx--be-a-polynomial-such-that--g2x--3--4x2--12x--5--find--gx2---1--94479092026-02-24T04:43:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--2x--3--then--x--racu---32--substitute-into--gu--4leftracu---32-7406542026-04-21T14:55:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-group-terms--9x2---2x---16y2--4y--144--complete-the-square--9x---12---1---16y--22---4--144--expand--9x---12---9---16y--22--64--144--simplify--9x---12---16y--22--89--the-center-is-at--1--2--final-answer-oxed-41830022026-02-24T04:43:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-must-be-divisible-by-the-least-common-multiple-lcm-of-7-11-and-13-since-these-are-all-primes-lcm--7--11--13--1001-the-smallest-four-digit-number-is-the-first-multiple-of-1001--1000-which-is-1001-imes-1--1001-thus-the-answer-is-oxed1001-69188272026-04-21T14:57:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-greatest-common-divisor-of-a-and-b-must-divide-their-sum-a--b--100-the-largest-divisor-of-100-is-50-to-achieve-gcda-b--50-set-a--50-and-b--50-thus-the-maximum-value-is-oxed50-1629762026-04-21T14:57:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-amount-of-saline-in-the-first-solution-be--020-times-3--06--liters-the-second-solution-contributes--050x--liters-of-saline-the-total-volume-is--3--x--liters-and-the-total-saline-is--0353--x--setting-up-the-equation-26401032026-04-21T15:06:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-model-states-that-each-hour-the-count-is-multiplied-by-3-after-1-hour--31--after-2-hours--32--so-after-4-hours--34--but-the-problem-states-the-population-is--36--times-initial-which-contradicts-unless-growth-rate-differs-however-assuming-the-model-is-correct-after-4-hours-32872622026-04-21T15:06:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--6--into-the-function-59006172026-04-21T15:06:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-local-extrema-of-the-function--pt--t3---6t2--9t--10--we-first-compute-its-derivative-90048592026-04-21T15:06:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-slant-asymptote-of--it--fract2--4t--2--we-perform-polynomial-long-division-of--t2--4--by--t--2--73170482026-04-21T15:07:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-of--ft--frac100tt2--4--we-compute-its-derivative-using-the-quotient-rule-52702712026-04-21T15:07:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-analyze-the-cubic-polynomial--rx--x3---3x--2--to-find-the-number-of-real-roots-we-first-attempt-rational-root-theorem-possible-rational-roots-are--pm1-pm2--testing-81964672026-04-21T15:07:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--iu--u---fracu39--and--ixn--frac1n--so-48408102026-04-21T15:07:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-standard-form-of-a-parabola-with-vertical-axis-is--y--frac14px2--the-focus-is-at--0-p--so--p--5--the-width-of-20-units-implies-the-parabola-passes-through--10-5--substituting--5--frac120100-rightarrow-5--5--which-holds-thus-the-equation-is--y--frac120x2--boxedy--dfrac120x2-53247762026-04-21T15:12:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-partial-fractions--frac1kk2--frac12left-frac1k---frac1k2-right--the-sum-becomes--frac12-left-sumk150-frac1k---sumk150-frac1k2-right--frac12-left-sumk150-frac1k---sumk352-frac1k-right--telescoping-gives--frac12-left-frac11--frac12---frac151---frac152-right--frac12-left-frac32---frac1032652-right--frac12-cdot-frac39752652--frac13251768--boxeddfrac13251768-19236732026-04-21T15:12:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-identity--p2--q2--p--q2---2pq--substituting--10---24i--3--4i2---2pq--compute--3--4i2--9--24i---16---7--24i--then--10---24i---7--24i---2pq--rearranging--17---48i---2pq-rightarrow-pq--frac-17--48i2--boxed-dfrac172--24i-44997962026-02-24T04:59:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-follows-a-binomial-distribution-with-parameters-n--4-trials-and-p--frac13-success-probability-the-formula-for-exactly-k-successes-is-binomnk-pk-1-pn-k-for-k--2-2758592026-04-21T15:15:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinct-elements-into-2-non-empty-subsets-which-is-the-stirling-number-of-the-second-kind-s6-2-the-formula-for-sn-2-is-2n-1---1-for-n--6-4005302026-04-21T15:15:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-binomial-distribution-with-n--3-p--frac14-and-k--1-74994492026-04-21T15:15:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-stars-and-bars-problem-the-formula-for-distributing-n-identical-items-into-k-distinct-boxes-is-binomn--k---1k---1-here-n--4-k--3-73814212026-04-21T15:15:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-numerator-and-denominator-by-the-conjugate-of-the-denominator-sqrt7---3-81569502026-04-21T15:15:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-terms-are-a-a--d-a--2d-a--3d-a--4d-20602452026-04-21T15:16:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation--x--y--4--describes-a-diamond-or-rhombus-centered-at-the-origin-to-find-the-vertices-consider-the-four-cases-for-absolute-values-50125262026-04-21T15:23:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-standard-form-of-the-ellipse-centered-at-the-origin-is-fracx2a2--fracy2b2--1-where-a--b-given-the-major-axis-is-10-we-have-2a--10-rightarrow-a--5-the-minor-axis-is-6-so-2b--6-rightarrow-b--3-the-distance-from-the-center-to-a-focus-is-given-by-c--sqrta2---b2-substituting-a--5-and-b--3-we-get-c--sqrt25---9--sqrt16--4-30212662026-04-21T15:53:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-maximize--ct--frac3tt2--4--we-take-its-derivative-using-the-quotient-rule-80906472026-04-21T15:53:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify-the-function-67612322026-04-21T15:53:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-between-two-unit-vectors--vecv--and--vecw--is-given-by-the-cosine-of-the-difference-of-their-angles-63554452026-04-21T15:53:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-parallelogram-formed-by-vectors--veca--and--vecb--is-the-magnitude-of-their-cross-product-59186662026-04-21T15:53:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--x--y--12--and--x2--y2--80--52566732026-04-21T15:57:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--a---b--6--and--a2--b2--130--55260532026-04-21T15:57:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--m--n--10---m2--n2--58--24329032026-04-24T12:42:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-lattice-points-on-the-hyperbola-fracx2144---fracy225--1-multiply-both-sides-by-144-cdot-25--3600-to-eliminate-denominators-89178332026-04-21T15:58:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-mathrmlcm18-24-36-77068172026-04-21T15:58:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-perimeter-of-a-rectangle-is-given-by--p--2l--w--where--l--is-length-and--w--is-width-given--2l--w--80--we-simplify-to--l--w--40--the-area--a--lw--express--w--40---l--so--a--l40---l--40l---l2--this-quadratic-equation-opens-downward-and-its-maximum-occurs-at-the-vertex--l--frac-b2a--frac-40-2--20--substituting--l--20---w--20--so--a--20-times-20--400--the-maximum-area-is-boxed400-58632352026-04-21T15:58:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-function--ft---5t2--30t--100--is-a-quadratic-opening-downward-the-vertex-occurs-at--t--frac-b2a--frac-302-5--3--substituting--t--3---f3---59--303--100---45--90--100--145--the-maximum-number-of-flowers-visited-is-boxed145-52938412026-04-21T15:58:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-group-and-complete-the-square-for--x--and--y--26839262026-04-21T15:58:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rearrange-the-equation--fx--y---fx--z--fy---fz--let--x--0---f-11730522026-04-21T15:58:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-paths-from-the-bottom-left-to-the-top-right-without-restrictions-the-robot-must-make-5-right-r-and-5-up-u-moves-so-62633882026-04-21T16:33:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-total-number-of-permutations-of-the-6-distinct-layers-83723312026-04-21T16:33:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-5-sensors-from-16-is-18262452026-04-21T16:33:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-8-digit-numbers-using-only-digits-3-and-4-30810942026-04-21T16:33:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-requires-distributing-5-distinct-keys-to-3-distinct-servers-with-each-server-receiving-at-least-one-key-this-is-a-classic-inclusion-exclusion-problem-the-total-number-of-ways-to-assign-the-keys-without-restriction-is-35-subtract-the-cases-where-at-least-one-server-gets-no-keys-binom31-cdot-25-add-back-the-cases-where-two-servers-get-no-keys-since-they-were-subtracted-twice-binom32-cdot-15-thus-the-total-is-98595582026-04-21T16:43:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-involves-arranging-7-earthquakes-where-3-are-identical-magnitude-75-and-4-are-identical-magnitude-68-the-number-of-distinct-sequences-is-the-multinomial-coefficient-5865812026-04-21T16:43:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-distinct-rankings-is-calculated-by-treating-the-61650132026-04-21T16:43:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-when-the-concentration-is-increasing-we-analyze-the-derivative-of--ct--using-the-quotient-rule-63249332026-04-21T16:46:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-vertical-asymptotes-and-holes-we-examine-the-denominator-of--dt--ract2--1t---2--59267472026-04-21T16:46:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate--f6--26596862026-04-21T16:47:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-terms-be--a---frac3d2-a---fracd2-a--fracd2-a--frac3d2--this-ensures-the-terms-are-in-arithmetic-progression-with-common-difference--d--the-first-term-is--a---frac3d2--and-the-last-term-is--a--frac3d2--their-sum-of-squares-is-22158242026-04-21T16:47:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-gcf-of-42-and-63-factor-both-numbers-42--2-times-3-times-7-and-63--32-times-7-the-common-prime-factors-are-3-and-7-multiply-them-3-times-7--21-boxed21-73050502026-04-21T16:55:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-simplifies-to-b---3--5-so-b---3--5-or-b---3---5-solving-gives-b--8-or-b---2-sum-8---2--6-boxed6-6020422026-02-24T06:46:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-increases-by-3-each-time-the-nth-term-is-2--n-1-times-3--3n---1-set-3n---1--32-3n--33-so-n--11-boxed11-question-1-45583762026-04-21T16:55:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-between-the-diagonal-of-a-square-and-its-side-is-45circ-but-here-the-angle-is-150circ-which-lies-in-the-second-quadrant-using-the-reference-angle-30circ-cos-150circ---cos-30circ---fracsqrt32-boxed-dfracsqrt32-659722026-02-24T07:02:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-with-sides-9-12-15-is-right-angled-since-92--122--152-the-area-is-frac12-cdot-9-cdot-12--54-the-altitudes-correspond-to-frac2-cdot-textareatextside-so-42069752026-04-21T17:07:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-volume-of-hemisphere-frac23pi-3x3--frac23pi-cdot-27x3--18pi-x3-39738952026-04-21T17:07:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-vertex-form-of-a-parabola--y--ax2--bx--c--has-its-vertex-at--x---fracb2a--here--a---2---b--8--so--x---frac82-2--2--substituting--x--2--into-the-equation--y---222--82---5---8--16---5--3--thus-the-vertex-is-at--2-3--boxed2-3-18071862026-02-24T07:03:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by-2--mv2--frac32kt--divide-by--m---v2--frac3kt2m--take-the-square-root--v--sqrtfrac3kt2m--boxedv--sqrtdfrac3kt2m-93426392026-04-21T17:07:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-maximum-height-occurs-at-the-vertex-for--y---x2--6x---8---a---1---b--6--so--x---frac62-1--3--substitute--x--3---y---32--63---8---9--18---8--1--the-maximum-height-is--1--unit-boxed1-73434922026-02-24T07:03:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--2x--1--so--x--fracu---12--substitute-into--fu---fu--4leftfracu---12right2--12leftfracu---12right--9--u---12--6u---1--9--u2---2u--1--6u---6--9--u2--4u--4--thus--fx2---3--x2---32--4x2---3--4--x4---6x2--9--4x2---12--4--x4---2x2--1--boxedx4---2x2--1-2358742026-04-21T17:07:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-1--frac2x---5x--3---1-geq-0-rightarrow-frac2x---5---x--3x--3-geq-0-rightarrow-fracx---8x--3-geq-0--critical-points-at--x--8--and--x---3--test-intervals-27288702026-02-24T07:03:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-factors-of-45-are-1-3-5-9-15-45-there-are-6-factors-the-probability-is-frac645--frac215-boxeddfrac215-94735272026-04-21T17:12:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-prime-factors-of-12-22-times-3-of-18-2-times-32-lcm-is-22-times-32--36-boxed36-82954042026-04-21T17:12:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-sum-is-5-times-22--110-sum-of-known-values-20--21--23--24--88-the-fifth-value-is-110---88--22-boxed22question-find-the-matrix-mathbfm-such-that-mathbfm-beginpmatrix-1--0-endpmatrix--beginpmatrix-2---1-endpmatrix-and-mathbfm-beginpmatrix-0--1-endpmatrix--beginpmatrix-3--4-endpmatrix-70874952026-02-24T07:08:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-matrix-mathbfm-is-constructed-by-placing-the-images-of-the-standard-basis-vectors-as-its-columns-thus-mathbfm--beginpmatrix-2--3---1--4-endpmatrix-verifying-mathbfm-beginpmatrix-1--0-endpmatrix--beginpmatrix-2---1-endpmatrix-and-mathbfm-beginpmatrix-0--1-endpmatrix--beginpmatrix-3--4-endpmatrix-confirming-correctness-boxedbeginpmatrix-2--3---1--4-endpmatrix-461612026-02-24T07:08:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-is-the-projection-of-4-3-onto-the-line-the-formula-for-the-projection-of-a-point-x0-y0-onto-ax--by--c--0-is-used-rewriting-the-line-as-frac12x--y---5--0-we-compute-the-projection-alternatively-parametrize-the-line-and-minimize-distance-let-x--t-then-y---frac12t--5-the-squared-distance-to-4-3-is-t---42--left-frac12t--5---3right2--t---42--left-frac12t--2right2-expanding-t2---8t--16--frac14t2---2t--4--frac54t2---10t--20-taking-derivative-and-setting-to-zero-frac52t---10--0-rightarrow-t--4-substituting-back-y---frac124--5--3-thus-the-closest-point-is-4-3-which-lies-on-the-line-boxed4-3-18029502026-04-21T17:12:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-theta-between-mathbfa-and-mathbfb-is-given-by-costheta--fracmathbfa-cdot-mathbfbmathbfa-mathbfb-compute-the-dot-product-21---34--2---12---10-compute-magnitudes-mathbfa--sqrt22---32--sqrt13-mathbfb--sqrt12--42--sqrt17-thus-costheta--frac-10sqrt13sqrt17-rationalizing-theta--arccosleft-frac10sqrt221right-boxedarccosleft-dfrac10sqrt221right-2891142026-04-21T17:12:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-scalar-triple-product-mathbfu-cdot-mathbfv-times-mathbfw-represents-the-volume-of-the-parallelepiped-formed-by-the-vectors-its-maximum-absolute-value-occurs-when-the-vectors-are-mutually-orthogonal-since-they-are-unit-vectors-the-maximum-value-is-1-cdot-1-cdot-1--1-for-example-if-mathbfu--mathbfv--mathbfw-are-orthonormal-the-53730152026-04-21T17:12:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-100-even-numbers-form-an-arithmetic-sequence-2--4--6--dots--200-the-sum-is-s--frac1002-times-2--200--50-times-202--10100-to-find-10100-mod-7-divide-10100-by-7-7-times-1442--10094-so-the-remainder-is-10100---10094--6-boxed6-68832002026-04-24T12:42:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x-be-the-integer-then-x-equiv--1-pmod5-and-x-equiv--1-pmod7-this-implies-x--1-is-divisible-by-both-5-and-7-so-x--1--textlcm5-7--35-thus-x--34-boxed34question-a-retired-engineer-designing-a-robotic-arm-observes-that-the-sum-of-two-gear-rotation-speeds--a--and--b--is-10-and-the-sum-of-their-squares-is-58-what-is--a3--b3--20751132026-02-24T07:15:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--a--b--10--and--a2--b2--58--we-want-to-find--a3--b3--89353622026-04-24T12:42:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initially-the-scientist-conducted-40-experiments-with-a-75-success-rate-24872962026-04-24T12:42:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--x--frac1x--4--square-both-sides-37690872026-04-22T06:24:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--sqrt7--2sqrt10--sqrt7---2sqrt10--we-compute--x2--39544532026-04-21T17:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-analyze-the-expression-by-considering-the-critical-points-where-expressions-inside-absolute-values-change-sign--x--frac52--and--x---3--we-consider-three-intervals-26448422026-04-21T17:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-sum-to-product-identity-66188352026-04-21T17:17:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-value-of--ft--t2---fract44--take-the-derivative--ft--2t---t3--set--ft--0---2t---t3--0-rightarrow-t2---t2--0--critical-points-at--t--0--and--t--pmsqrt2--evaluate--fsqrt2--sqrt22---fracsqrt244--2---frac44--1--the-maximum-value-is--boxed1--58703972026-04-21T17:24:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--b--10--and--a2--b2--58--compute--a--b2--100--a2--2ab--b2--58--2ab--solve-for--ab---2ab--42-rightarrow-ab--21--the-area-is--frac12ab--boxedfrac212--96172082026-04-21T17:24:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify--pn--fracnnn2--2--frac1n2--2--as--n-to-infty---n2--2-to-infty--so--limn-to-infty-pn--boxed0--79543352026-04-21T17:24:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-take-derivative--ax---frac4x2---2x--set--ax--0----frac4x2---2x--0-rightarrow--4---2x3--0-rightarrow-x3---2--no-positive-roots-analyze-behavior-as--x-to-0---a-to-infty--as--x-to-infty---a-to--infty--thus-no-maximum-exists-the-function-increases-without-bound-near--x--0--however-if-constrained-recheck-calculations-correct-approach-13958122026-02-24T07:32:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-count-of-integers--n--such-that--1-leq-n-leq-500--and--n-equiv-2-pmod7--we-note-the-sequence-starts-at-2-and-increases-by-7-each-time--2-9-16-dots--the-general-term-is--7k--2--solving--7k--2-leq-500--gives--k-leq-rac4987-pprox-7114--so--k--0-1-dots-71--this-yields--72--integers-oxed72-48800582026-04-21T17:44:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-divisors-of-120-are--1-2-3-4-5-6-8-10-12-15-20-24-30-40-60-120--those-divisible-by-5-are--5-10-15-20-30-40-60-120--summing-them--5--10--15--20--30--40--60--120--290--oxed290question-let--x--and--y--be-positive-real-numbers-representing-the-bandwidths-in-gbps-of-two-parallel-data-streams-find-the-minimum-value-of-271372026-04-21T17:44:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--racx2--y2xy--and--b--racxyx2--y2--then-the-expression-becomes-85215592026-04-21T17:44:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--racp--qp---q--and--b--racp---qp--q--note-that--b--rac1a--and-since--p--q--0---a--1--then-40823692026-04-21T17:44:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-hours-in-a-week-is-7-times-24--168-the-prime-factorization-of-168-is-23-times-3-times-7-the-odd-divisors-are-the-divisors-of-3-times-7--21-which-are-1-3-7-21-their-sum-is-1--3--7--21--32-17775722026-04-21T18:23:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-word-atmospheric-has-4-vowels-a-o-a-i-and-mitigation-has-3-vowels-i-i-i-the-gcd-of-4-and-3-is-1-since-they-share-no-common-divisors-other-than-1-63454082026-04-21T18:23:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-word-climate-has-7-letters-and-modeling-has-8-letters-the-lcm-of-7-and-8-is-56-the-smallest-three-digit-multiple-of-56-is-112-since-56-times-2--112-10036702026-04-21T18:23:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-time-between-two-consecutive-orbits-is--frac5pi2--hours-since--pi-approx-31416--we-compute-33728592026-04-21T18:39:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-birds-follows-a-geometric-sequence--3-6-12-24-dots--with-first-term--a--3--and-common-ratio--r--2--57631932026-04-21T18:39:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--rt--sqrtt---82--78198332026-04-21T18:39:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-total-cost-for-two-sensors-is-88-so-each-sensor-costs-62431232026-04-21T18:39:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-tasked-with-finding-the-number-of-integer-solutions-x-y-to-the-equation--x2---y2--2025--this-can-be-factored-as-17390102026-04-21T18:40:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor--1440--76053832026-04-21T18:40:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-an-arithmetic-series-of-odd-numbers-from-1001-to-1099-inclusive-46889592026-04-21T18:40:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-probability-problem-let--x--xs-xp-xf--be-the-counts-of-each-outcome-in-12-independent-trials-with-probabilities--ps--04---pp--03--and--pf--03--the-probability-mass-function-of-a-multinomial-distribution-gives-92951392026-04-21T18:44:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-4-instruments-from-10-is-85299012026-04-21T18:44:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-probability-let--x--xa-xb-xc--be-the-counts-of-each-letter-in-a-random-8-character-word-with-each-character-independently-chosen-from-a-b-c-with-equal-probability--rac13--85184532026-04-21T18:44:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-distribution-with-10-independent-trials-and-3-categories-lets-define-the-counts-based-on-the-actual-route-types-92361482026-04-21T18:44:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--fx--is-a-polynomial-and-four-values-are-given-and-the-behavior-resembles-a-quadratic-pattern--3-8-15-24--we-first-examine-the-second-differences-20589392026-04-21T19:05:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-functional-equation-is--fxy--fx-y--2fx--2fy--this-is-a-known-variant-of-the-quadratic-functional-equation-19288262026-04-21T19:06:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--x2---1--then--x2--y--1--substitute-into--gy--2x4---5x2--1--61251102026-04-21T19:26:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--x2--2--then--x2--y---2--express--gy--in-terms-of--y--73572442026-04-21T19:26:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-point-on-the-line--y--frac34x--2--closest-to--0-0--we-minimize-the-square-of-the-distance-function-14197152026-04-21T19:26:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-expand--sin-x--cos-x2--sin2-x--2sin-x-cos-x--cos2-x--1--2sin-x-cos-x--30649722026-04-21T19:27:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--mathbfa-mathbfb-mathbfc--are-unit-vectors-we-use-the-identity-24223732026-04-21T19:27:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-given-points-be--a--111--76110462026-04-21T19:27:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--fx--is-a-cubic-polynomial-such-that-38642042026-04-21T19:34:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--dt--is-a-cubic-polynomial-satisfying-53805912026-04-21T19:34:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-we-first-identify-the-positive-integers-less-than-30-that-are-divisible-by-both-3-and-5-these-numbers-are-multiples-of-the-least-common-multiple-of-3-and-5-which-is-15-the-multiples-of-15-less-than-30-are-90183562026-04-21T19:37:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-find-all-positive-integers-less-than-or-equal-to-50-that-are-factors-of-60-the-positive-divisors-of-60-are-93128082026-04-21T19:37:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-length-is-calculated-by-adding-the-two-lengths-and-dividing-by-2-60037462026-04-21T19:37:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-prime-factor-of-1211-we-test-divisibility-by-small-primes-first-1211-is-odd-so-not-divisible-by-2-the-sum-of-the-digits-is--1--2--1--1--5--not-divisible-by-3-it-doesnt-end-in-0-or-5-so-not-divisible-by-5-next-test-divisibility-by-7-68414802026-04-21T19:37:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-a-vector-orthogonal-to-both-compute-their-cross-product-64285352026-04-21T20:19:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-binomial-probability-problem-with--n--7---p--04--and-we-want--px-geq-5--px5--px6--px7--76638442026-04-21T20:19:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-binomial-probability-problem-with--n--6---p--03--and-we-want--px--3--px0--px1--px2--46525652026-04-21T20:19:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-circular-arrangements-of--n--distinguishable-people-the-number-of-distinct-arrangements-is--n-1--since-rotations-are-equivalent-96273982026-04-21T20:19:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-hypergeometric-probability-problem-8277902026-04-21T20:19:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-smallest-two-digit-positive-integer--l--such-that-71537522026-04-21T20:31:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-sum-the-values-and-divide-by-the-number-of-days-84603212026-04-21T20:36:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-square-root-of-a-square-simplifies-to-the-absolute-value-3124802026-04-21T20:36:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-prime-factorize-both-numbers-40571382026-04-21T20:36:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-simplify-the-equation-using-absolute-value-17391352026-04-21T20:36:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-for--hx2---2--start-with-the-given-function-8488792026-04-21T20:50:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-and-width-of-the-rectangle-be--l--and--w--respectively-the-perimeter-is-given-by-84102902026-04-21T20:50:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-d--gcda-b-then-we-can-write-a--d-cdot-m-and-b--d-cdot-n-where-m-and-n-are-coprime-positive-integers-the-total-road-length-is-a--b--dm--n--1000-so-d-must-divide-1000-to-maximize-d-we-minimize-m--n-subject-to-m-and-n-being-coprime-positive-integers-the-smallest-possible-value-of-m--n-is-2-which-occurs-when-m--n--1-and-they-are-coprime-this-gives-d--frac10002--500-since-m--1-and-n--1-are-coprime-this-is-valid-therefore-the-largest-possible-value-of-gcda-b-is-boxed500-21267542026-04-21T20:59:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-least-common-multiple-lcm-of-18-and-30-first-factor-both-numbers-55248222026-04-21T20:59:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-d--gcda-b-so-a--d-cdot-m-b--d-cdot-n-with-m-and-n-coprime-positive-integers-then-61718352026-04-21T20:59:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-d--gcda-b-so-a--d-cdot-m-b--d-cdot-n-with-m-and-n-coprime-then-80545182026-04-21T20:59:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-sides-be-a-1-a-a1-this-is-a-triangle-only-if-triangle-inequalities-hold-48787782026-04-21T20:59:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute--g3--4i--3--4i--since--gz--returns--z--itself-then--fg3--4i--f3--4i--3--4i2--9--24i--16i2--9--24i---16---7--24i--the-result-is-boxed-7--24i-74317372026-04-21T21:12:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--alpha--sqrt1--i---beta--sqrt1---i--the-conjugate-pairs--alpha--and---alpha---beta--and---beta--must-both-be-roots-for-real-coefficients-but-since-the-polynomial-is-monic-of-degree-2-and-has-only-these-two-specified-roots-we-must-consider-symmetry-instead-compute-the-sum-and-product-note--1--i--1---i--2--and--1--i1---i--1--1--2--let--z2----alpha--beta-z--alphabeta--but-observing-that--alphabeta--sqrt1i1-i--sqrt2--also--alpha2--beta2--2--and--alpha2beta2--2--let--s--alpha--beta--then--s2--alpha2--beta2--2alphabeta--2--2sqrt2--but-to-find-a-real-polynomial-consider-that--alpha--sqrt1i--and--sqrt1i--sqrtsqrt2-eipi8--214-cos-fracpi8--isin-fracpi8--however-instead-of-direct-polar-form-consider-squaring-the-sum-alternatively-note-that--alpha--and--beta--are-conjugate-like-in-structure-but-realize--sqrt1i--and--sqrt1-i--are-not-conjugates-but-if-we-form-a-polynomial-with-both-and-require-real-coefficients-then-the-minimal-monic-polynomial-must-have-roots--sqrt1i--sqrt1i-sqrt1-i--sqrt1-i--unless-paired-but-the-problem-says-roots-at-these-two-so-assume--alpha--sqrt1i---beta--sqrt1-i--and-for-real-coefficients-must-include---alpha--beta--but-that-gives-four-roots-therefore-likely-the-polynomial-has-roots--sqrt1i--and--sqrt1-i--and-since-coefficients-are-real-it-must-be-invariant-under-conjugation-but--overlinesqrt1i--sqrt1---i--beta--so-if--alpha--sqrt1i--then--overlinealpha--beta--thus-the-roots-are--alpha--and--overlinealpha--so-the-monic-quadratic-is--z---alphaz---overlinealpha--z2---2operatornamerealpha-z--alpha2--now--alpha2--1i--so--alpha2--alpha2--1i--sqrt2--also--2operatornamerealpha--alpha--overlinealpha--but--alpha--overlinealpha2--alpha2--2alpha2--overlinealpha2--wait-better--operatornamerealpha--frac-alpha--overlinealpha-2--from--alpha2--1i--take-real-part--operatornamerealpha2--operatornamere1i--1--alpha2-cos2theta---operatornameimalpha2--sin2theta--1--so--cos2theta--1sqrt2---sin2theta--1sqrt2--so--2theta--pi4---theta--pi8--then--operatornamerealpha--alpha-costheta--sqrt2-cospi8--but--cospi8--sqrt2--sqrt2--2--so--operatornamerealpha--sqrt2-cdot-frac-sqrt2--sqrt2-2--frac-sqrt2-sqrt2--sqrt2-2--this-is-messy-instead-use-identity--alpha2--1i--so--alpha4--1i2--2i--but-for-the-polynomial--z---alphaz---beta--z2---alpha--betaz--alphabeta--note--alphabeta--sqrt1i1-i--sqrt2--now--alpha--beta2--alpha2--beta2--2alphabeta--1i--1-i--2sqrt2--2--2sqrt2--so--alpha--beta--sqrt2--2sqrt2--but-this-is-not-helpful-note--alpha--and--beta--satisfy-a-polynomial-whose-coefficients-are-symmetric-but-recall-the-minimal-monic-polynomial-with-real-coefficients-having--sqrt1i--as-a-root-must-also-have---sqrt1i--unless-we-accept-complex-coefficients-but-we-want-real-so-likely-the-intended-polynomial-is-formed-by-squaring-suppose--z--sqrt1i--then--z2---1--i--so--z2---12---1--so--z4---2z2--1---1-rightarrow-z4---2z2--2--0--but-this-has-roots--pmsqrt1i-pmsqrt1-i--check-if--z2--1i---z4---2z2--2--1i2---21i--2--12i-1--2--2i--2--0--2i---2i--0--0-wait--1i2--1--2i--1--2i--then--2i---21i--2--2i--2--2i--2--0--yes-so--z4---2z2--2--0--has-roots--pmsqrt1i-pmsqrt1-i--but-the-problem-wants-a-quadratic-however-if-we-take--z--sqrt1i--and---sqrt1-i--no-but-notice-the-root--sqrt1i--and-its-negative-is-also-a-root-if-polynomial-is-even-but--f-z--fz--only-if-symmetric-but--fz--z2---1---i--has--sqrt1i--but-not-symmetric-the-minimal-real-coefficient-polynomial-with--sqrt1i--as-root-is-degree-4-but-the-problem-likely-intends-the-monic-quadratic-formed-by--sqrt1i--and-its-conjugate--sqrt1-i--even-though-it-doesnt-have-real-coefficients-unless-paired-but--sqrt1-i--is-not---overlinesqrt1i--let--alpha--sqrt1i---overlinealpha--sqrt1-i--since--overlinesqrt1i--sqrt1-overlinei--sqrt1-i--yes-complex-conjugation-commutes-with-square-root-only-if-domain-is-fixed-but--overlinesqrtz--sqrtoverlinez--for--overlinez--in-domain-of-definition-assuming--sqrt1i--is-taken-with-positive-real-part-then--overlinesqrt1i--sqrt1-i--so-the-conjugate-is--sqrt1-i--overlinealpha--so-for-a-polynomial-with-real-coefficients-if--alpha--is-a-root-so-is--overlinealpha--so-the-roots-are--sqrt1i--and--sqrt1-i--overlinesqrt1i--therefore-the-monic-quadratic-is--z---sqrt1iz---overlinesqrt1i--z2---2operatornameresqrt1i-z--sqrt1i2--now--sqrt1i2--alpha2--1i--sqrt2--no--alpha2--alpha2--1i--sqrt2--no--alpha2---alpha2-1--no--alpha2--alpha2--and--alpha2--1i--so--alpha2--1i--sqrt12--12--sqrt2--yes-and--operatornamerealpha--frac-alpha--overlinealpha-2--from--alpha2--1i--take-modulus--alpha4--1i2--2--so--alpha22--2--thus--alpha4--2--so--alpha2--sqrt2--since-magnitude-positive-so--operatornamerealpha--frac-alpha--overlinealpha-2--but--alpha--overlinealpha2--alpha2--2alpha2--overlinealpha2--no--overlinealpha2--overlinealpha2--overline1i--1-i--so--alpha--overlinealpha2--alpha2--2alphaoverlinealpha--overlinealpha2--1i--1-i--2alpha2--2--2sqrt2--therefore--alpha--overlinealpha--sqrt2--2sqrt2--so-the-quadratic-is--z2---sqrt2--2sqrt2--z--sqrt2--but-this-is-not-nice-wait--theres-a-better-way-note-that--sqrt1i--fracsqrt221i12--but-perhaps-the-intended-answer-is-to-use-the-identity-the-polynomial-whose-roots-are--sqrt1pm-i--is--z4---2z2--2--0--but-we-want-quadratic-but-the-only-monic-quadratic-with-real-coefficients-having--sqrt1i--as-a-root-must-also-have---sqrt1i---overlinesqrt1i----overlinesqrt1i--and-if-its-degree-4-but-the-problem-asks-for-quadratic-unless--sqrt1i--is-such-that-its-minimal-polynomial-is-quadratic-but-its-not-as--mathbbqsqrt1imathbbq--4--but-perhaps-in-the-context-they-want--z---sqrt1iz---sqrt1-i--but-again-not-real-after-reconsideration-the-intended-solution-likely-assumes-that-the-conjugate-is-included-and-the-polynomial-is--z2---2cospi8sqrt2-z--sqrt2--but-thats-not-nice-alternatively-recognize-that--1i--sqrt2-eipi4--so--sqrt1i-57287542026-02-24T13:08:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--f2--g2--compute--f2--4---6--m---2--m--and--g2--4---6--5m---2--5m--equate---2--m---2--5m--simplify--0--4m--so--m--boxed0--25365842026-02-24T13:44:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-identity--a3--b3--a--b3---3aba--b--first-find--ab---a--b2--a2--2ab--b2-rightarrow-49--35--2ab-rightarrow-ab--7--then--a3--b3--343---3-cdot-7-cdot-7--343---147--boxed196--32278062026-04-21T21:35:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-from-the-second-equation--x--y--2--substitute-into-the-first--2y--2--y--10-rightarrow-2y--4--y--10-rightarrow-3y--6-rightarrow-y--2--then--x--2--2--4--the-intersection-is--boxed4-2--72054462026-04-21T21:35:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides--x2--2--frac1x2--25-rightarrow-x2--frac1x2--23--multiply-by-2--2x2--frac2x2--46--final-answer--boxed46-question-in-a-sustainable-desert-agriculture-project-the-combined-water-efficiency-of-two-systems-is-modeled-by--a--b--10--and--a2--b2--58--determine-the-total-water-efficiency-cubed--a3--b3--8987202026-04-21T21:35:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--fracab--3--let--a--3b--substituting-into--a--b--16---3b--b--16--so--b--4--and--a--12--now--a2---b2--factors-as--a---ba--b--12---416--8-times-16--128--therefore-the-value-is--boxed128--77625722026-04-21T21:35:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-how-many-of-the-first-100-positive-integers-satisfy-the-congruence-80320842026-04-21T22:08:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-least-common-multiple-of-5-and-7-61317972026-04-21T22:08:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-two-digit-number-36842192026-04-21T22:08:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-100th-signal-occurs-at-minute-7319082026-04-21T22:08:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-divisibility-rule-for-9-the-remainder-when-a-number-is-divided-by-9-is-the-same-as-the-remainder-when-the-sum-of-its-digits-is-divided-by-9-94474572026-04-21T22:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-euclidean-algorithm-33789592026-04-21T22:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfm--beginpmatrix-a--b--c--d-endpmatrix-multiply-mathbfm-with-beginpmatrix-1--2--3--4-endpmatrix-to-get-35704152026-04-21T22:42:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-pythagorean-theorem-since-overrightarrowoa-and-overrightarrowob-are-perpendicular-compute-overrightarrowoc2--2overrightarrowoa2--overrightarrowob2--425--144--100--144--244-thus-overrightarrowoc--sqrt244--2sqrt61-final-answer-boxed2sqrt61-29897952026-04-21T22:43:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-system-84960962026-04-21T22:43:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-define-a-function--fu--such-that-39483672026-04-21T22:54:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--gx--is-a-cubic-polynomial-53311112026-04-21T23:22:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-functional-equation-is-87836522026-04-21T23:22:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--px--is-a-cubic-polynomial-when-dividing-by--x4---1--the-remainder--rx--must-be-a-polynomial-of-degree-less-than-4-however-because--x4---1--has-roots--pm-1-pm-i--and--px--is-real-the-remainder-can-be-expressed-as-any-cubic-or-lower-but-here-we-are-to-find-the-actual-remainder-which-is-unique-and-of-degree---4--but-since--px--is-already-degree-3-and-division-by-degree-4-polynomial-yields-a-unique-remainder-of-degree---4--and-since--px--agrees-with-any-degree-3-polynomial-at-these-4-points-we-conclude-32633612026-04-21T23:22:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-apply-the-cauchy-schwarz-inequality-in-the-engel-form-also-known-as-titus-lemma-38158442026-04-21T23:22:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-value-of--k--substitute--x--3--into-the-function-and-set--f3--0--2241972026-04-22T00:16:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-population-stabilizes-at--t---6--this-means--p-6--0--substituting-into-the-equation-38543752026-04-22T00:16:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-rationalize-the-denominator-of--frac3x--4sqrtx---2--multiply-numerator-and-denominator-by-the-conjugate--sqrtx--2--39736262026-04-22T00:16:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression--x2--5x--6--is-a-quadratic-trinomial-we-look-for-two-numbers-that-multiply-to-6-and-add-to-5-these-numbers-are-2-and-3-3765522026-04-22T00:16:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-point--3-11--lies-on-the-line--y--2x--b--substitute--x--3--and--y--11--99466622026-04-22T00:16:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-vertex-of-a-parabola--hx--x2---4x--c--occurs-at--x--frac--421--2--which-matches-the-given-condition-now-substitute--x--2--and-set--h2--3--18341152026-04-22T00:16:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--fx--ax3--bx2--cx--d--we-use-the-given-values-to-form-a-system-of-equations-86098702026-04-22T00:16:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--a2--b2--1--and--ab--frac12--we-are-to-find--a4--b4--56525522026-04-22T00:41:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-minimize-11188412026-04-22T00:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--sqrtu--t--so--u--t2--substituting-into-the-equation-45656372026-04-22T00:41:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-spherical-coordinates-phi-is-the-polar-angle-measured-from-the-positive-z-axis-when-phi-is-constant-the-set-of-points-forms-a-cone-with-its-apex-at-the-origin-opening-symmetrically-around-the-z-axis-for-phi--fracpi4-the-cone-makes-a-45circ-angle-with-the-z-axis-thus-the-shape-is-a-cone-88410632026-04-22T00:42:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-theta-be-the-angle-between-any-pair-of-vectors-since-mathbfu-cdot-mathbfv--costheta--frac12-theta--60circ-the-magnitude-squared-is-mathbfu--mathbfv--mathbfw2--3--2mathbfu-cdot-mathbfv--mathbfu-cdot-mathbfw--mathbfv-cdot-mathbfw--3--23-cdot-frac12--3--3--6-thus-the-maximum-value-is-sqrt6-58035712026-04-22T00:42:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-orthogonality-their-dot-product-must-be-zero-1-cdot-x--x-cdot-3--2-cdot--1--0-simplifying-x--3x---2--0-rightarrow-4x--2-rightarrow-x--frac12-21482872026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-angle-aob--90circ-overrightarrowoa-cdot-overrightarrowob--0-compute-overrightarrowoc2--4overrightarrowoa2--overrightarrowob2---4overrightarrowoa-cdot-overrightarrowob--44--9---0--25-thus-overrightarrowoc--5-21669612026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-t--sin-x--csc-x-note-tan-x--cot-x--fracsin-xcos-x--fraccos-xsin-x--frac1sin-x-cos-x-let-y--sin-x-cos-x--frac12sin-2x-so-tan-x--cot-x--frac2sin-2x-the-expression-becomes-leftfrac2sin-2xright2--sin-x--csc-x2-let-z--sin-x--csc-x-geq-2-by-am-gm-substitute-u--sin-2x-then-minimize-frac4u2--leftfrac1sin-x--sin-xright2-after-calculus-or-substitution-the-minimum-occurs-at-x--fracpi4-yielding-1--12--sqrt2--frac1sqrt22--4--leftfrac3sqrt2right2--4--frac92--frac172-however-correct-simplification-shows-the-minimum-is-9-when-x--fracpi4-94555132026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-integer-be--n--we-have--n-equiv-0-pmod16--and--n-equiv-1-pmod5--let--n--16k--substituting-into-the-second-congruence-37291022026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-divisors-of--12--are--1-2-3-4-6-12--compute-the-sum-of-their-squares-96423612026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-product-of--k--consecutive-integers-is-divisible-by--k--for-five-consecutive-integers-this-is--5--120--for-example--1-cdot-2-cdot-3-cdot-4-cdot-5--120--and--2-cdot-3-cdot-4-cdot-5-cdot-6--47554192026-04-22T00:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-surface-area-of-a-sphere-is--4pi-r2--for-the-pollen-grain-this-is--4pi-r2--the-surface-area-of-a-hemisphere-includes-the-curved-surface--2pi-3r2--and-the-flat-circular-base--pi-3r2--so-total-surface-area-is--2pi-9r2--pi-9r2--18pi-r2--9pi-r2--27pi-r2--the-ratio-is-86868522026-04-22T00:45:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circumscribed-circle-using-the-pythagorean-theorem-9079762026-04-22T00:45:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-cone-is--frac13pi-r2-h--and-the-volume-of-a-sphere-is--frac43pi-r3--setting-them-equal-94081992026-04-22T00:45:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-circle-has-radius--5--micrometers-so-its-area-is-41684482026-04-22T00:45:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-volume-of-the-sphere-46871062026-04-22T00:45:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-inscribed-to-plain-tablets-is-53-so-the-total-number-of-parts-is-5--3--8-since-there-are-48-tablets-each-part-represents--frac488--6--tablets-thus-the-number-of-inscribed-tablets-is-5-times-6--30-we-are-told-that-inscribed-tablets-must-be-displayed-in-groups-of-7-so-we-seek-the-greatest-multiple-of-7-that-is-less-than-or-equal-to-30-the-multiples-of-7-below-30-are-7-14-21-28-the-greatest-is-28-therefore-the-largest-number-of-inscribed-tablets-that-can-be-grouped-in-sevens-is-28-39714752026-04-22T00:45:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-emissions-is-479-let-the-common-multiplier-be-x-then-the-second-borough-emitted-7x--147-metric-tons-solving-for-x-47661242026-04-22T00:45:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-area-of-the-hall-is-12-times-12--144-square-meters-each-rectangle-has-area-at-least-18-so-the-maximum-number-of-rectangles-is--leftlfloor-frac14418-rightrfloor--8--but-we-want-the-minimum-number-so-we-seek-the-largest-rectangle-area-that-divides-144-and-is--geq-18--and-check-if-tiling-with-such-rectangles-covers-the-square-34770562026-04-22T00:45:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-total-be-1-add-the-fractions-of-coniferous-and-deciduous-86612992026-04-22T00:46:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-five-values-is--frac58--so-the-total-rainfall-is-76326292026-04-22T00:46:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-sum--sumk16-k2--12--22--32--42--52--62--1--4--9--16--25--36--91--next-compute-the-product--prodj13-2j--1--2cdot1--12cdot2--12cdot3--1--3-cdot-5-cdot-7--105--now-multiply-the-results--91-times-105--9105--therefore-the-value-is--boxed9105--81801362026-04-22T00:46:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-slope--m--of-the-line-is--frac19---79---3--frac126--2--using-point-slope-form-with-point--3-7---y---7--2x---3--simplify--y--2x---6--7--2x--1--the--y--intercept-occurs-when--x--0--so--y--20--1--1--thus-the--y--intercept-is--boxed1--99240122026-04-22T00:46:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--x2--so-the-expression-becomes--u2---13u--36--factor--u---9u---4--substitute-back--x2---9x2---4--each-is-a-difference-of-squares--x---3x--3x---2-8260192026-04-22T00:46:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-dot-product-of-two-unit-vectors-is-mathbfu-cdot-mathbfv--costheta-given-costheta--fracsqrt32-the-angle-theta-satisfies-theta--arccosleftfracsqrt32right-this-corresponds-to-theta--30circ-or-fracpi6-radians-however-since-cosine-is-positive-in-both-the-first-and-fourth-quadrants-but-angles-between-vectors-are-typically-taken-in-0-pi-the-solution-is-boxeddfracpi6-16579722026-04-22T00:59:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-cosine-of-the-angle-between-vectors-mathbfa-and-mathbfb-is-given-by-costheta--fracmathbfa-cdot-mathbfbmathbfa-mathbfb-compute-the-dot-product-mathbfa-cdot-mathbfb--10--01--11--1-the-magnitudes-are-mathbfa--sqrt12--02--12--sqrt2-and-mathbfb--sqrt02--12--12--sqrt2-thus-costheta--frac1sqrt2-cdot-sqrt2--frac12-the-final-answer-is-boxeddfrac12-27948272026-04-22T00:59:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-magnitude-of-the-cross-product-is-mathbfp-times-mathbfq--mathbfp-mathbfq-sintheta-since-mathbfp-and-mathbfq-are-unit-vectors-mathbfp-times-mathbfq--sintheta--frac12-thus-theta--arcsinleftfrac12right-which-gives-theta--fracpi6-or-frac5pi6-however-the-angle-between-vectors-is-taken-as-the-acute-angle-so-theta--boxeddfracpi6-60680422026-04-22T00:59:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero-mathbfm-cdot-mathbfn--x1--2-1--x---2--0-solving-gives-x--2-the-final-answer-is-boxed2-68159382026-02-24T18:42:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-w--z2-so-w2--w--1--0-solving-gives-w--frac-1-pm-sqrt-32--epm-2pi-i3-thus-z2--epm-2pi-i3-so-z--epm-pi-i3--pi-i-k2-for-k--0-1-the-roots-have-angles-pi3-5pi3-7pi3-11pi3-but-68374072026-04-22T00:59:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-hemisphere-is--frac23pi-3x3--frac23pi-27x3--18pi-x3--the-cylinders-volume-is--pi-x2-cdot-3x--3pi-x3--the-ratio-is--frac18pi-x33pi-x3--boxed6--70587272026-04-22T01:02:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-radius--r--sqrtfrac100pipi--10--textkm--new-radius--r--8--textkm--original-area--100pi--new-area--pi-82--64pi--the-decrease-is--100pi---64pi--boxed36pi--textkm2--47521532026-04-22T01:02:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-the-inradius--r--fraca--b---c2--where--c--is-the-hypotenuse-let--a--and--b--be-the-legs-then--r--fraca--b---252--6-rightarrow-a--b--37--the-area-of-the-triangle-is--fracab2--and-the-inradius-formula--r--fraca--b---c2--gives-consistency-using--a2--b2--625--and--a--b--37--solve-for--ab---a--b2--1369--a2--b2--2ab--625--2ab-rightarrow-ab--372--area---186--textm2--circle-area--pi-r2--36pi--ratio--frac36pi186--boxedfrac6pi31--82799222026-04-22T01:02:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-slope-forms-a-right-triangle-with-opposite-side-1-and-adjacent-side--sqrt3--thus--tan-theta--frac1sqrt3--boxedfracsqrt33-question-what-is-the-remainder-when-the-sum--13--23--33--dots--103--is-divided-by-9-39786032026-04-22T01:02:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-formula-for-the-sum-of-the-first--n--cubes-is--left-fracnn12-right2--for--n--10--we-compute-75392042026-04-22T01:02:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--3217--to-base-ten-expand-using-powers-of-7-2509672026-04-22T01:02:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-gcd-take-the-minimum-exponent-for-each-common-prime-factor-86724412026-04-22T01:02:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-lcm-take-the-maximum-exponent-for-each-prime-factor-6130782026-04-22T01:02:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-the-divisors-of--23-cdot-32--is-given-by-74266752026-04-22T01:02:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-note-that--7-equiv-1-mod-6--so--7k-equiv-1k--1-mod-6--for-any-positive-integer--k--2167012026-04-22T01:02:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-prime-factorization-is-already-given--24-cdot-33-cdot-52--the-largest-prime-factor-is-clearly--boxed5--63744302026-04-22T01:02:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-of-12-and-18-factor-both-9355352026-04-22T01:02:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-84-and-126-factor-both-numbers-51495732026-04-22T01:06:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-the-sum-divided-by-3-545212026-04-22T01:06:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-microscopes-be-x-the-ratio-54--15x-implies-57752322026-04-22T01:06:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-expressions-69068492026-04-22T01:06:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term-a1--2-and-common-difference-d--3-the-n-th-term-is-79054672026-04-22T01:06:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-original-side-length-of-the-square-is--frac404--10--textcm--the-original-area-is--102--100--textcm2--after-increasing-each-side-by--2--textcm--the-new-side-length-is--12--textcm--and-the-new-area-is--122--144--textcm2--the-increase-in-area-is--144---100--44--textcm2--20981642026-04-22T01:06:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-cosine-of--360circ--corresponds-to-the-x-coordinate-of-the-point-on-the-unit-circle-at-a-full-rotation-from-the-positive-x-axis-this-point-is--1-0--so--cos-360circ--1--51920292026-04-22T01:06:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-cylinder-is--vtextcyl--pi-r2-6r--6pi-r3--the-volume-of-the-cone-is--vtextcone--frac13-pi-2r2-3r--frac13-pi-4r23r--4pi-r3--the-ratio-is-72837642026-04-22T01:06:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-herons-formula-the-semi-perimeter-is-14946292026-04-22T01:06:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-surface-area-of-a-cone-is-the-sum-of-the-base-area-and-the-lateral-surface-area-the-base-area-is-20227442026-04-22T01:06:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-tangent-of--45circ--is-defined-as--fracsin-45circcos-45circ--fracfracsqrt22fracsqrt22--1--31757652026-04-22T01:06:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diameter-of-the-inscribed-circle-is-equal-to-the-side-length-of-the-square-which-is--8--textcm--therefore-the-radius-is--frac82--4--textcm--the-area-of-the-circle-is-8030782026-04-22T01:06:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-side-length-of-the-hexagon-is--frac366--6--text-91164762026-04-22T01:06:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-is-given-by--v--frac43-pi-r3--differentiating-with-respect-to-time--t--we-get-26750832026-04-22T01:12:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-has-side-lengths--a--13---b--14--and--c--15--first-compute-the-semi-perimeter--s--25266132026-04-22T01:12:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-original-side-length-of-the-equilateral-triangle-is--frac363--12--cm-the-area-of-an-equilateral-triangle-is-given-by-20333892026-04-22T01:12:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-hypotenuse-be--z--25--m-and-the-inradius-be--c--5--m-for-a-right-triangle-the-area--a--is-given-by-36250682026-04-22T01:12:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--3--into--fx--92296262026-04-22T01:32:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--5--into--hx--39484692026-04-22T01:32:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--y--x2---3x--then-the-product-becomes--y--2my--5m--y2--7my--10m2--substitute-back-55070392026-04-22T01:32:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--m---3--83281692026-04-22T01:32:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-liters-of-15-fertilizer-and--y--be-the-liters-of-40-fertilizer-the-system-is-53881552026-04-22T01:32:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-be--5k--and-height--3k--by-the-pythagorean-theorem-51102612026-04-22T01:32:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-of-neither-test-being-positive-is--left1---rac23-84758032026-04-22T01:32:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with-a1--1-compute-a2--ma1--1---rac122--rac12-next-compute-a3--ma2--rac12---racrac1222--rac12---rac18--rac38-thus-the-value-of-a3-is-oxeddfrac38-75620972026-04-22T01:32:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-cauchy-schwarz-inequality-22--32--42x2--y2--z2-geq-2x--3y--4z2-this-gives-29x2--y2--z2-geq-144-so-x2--y2--z2-geq-rac14429-equality-holds-when-racx2--racy3--racz4--k-leading-to-x--2k-y--3k-z--4k-substituting-into-2x--3y--4z--12-gives-4k--9k--16k--29k--12-so-k--rac1229-thus-the-minimum-value-is-oxeddfrac14429-21116062026-04-22T01:32:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-recursive-sequence-defined-by--bn1--gbn--where--gt--t---ract24--and--b1--1--we-are-to-find--limn-o-infty-bn--assuming-the-limit-exists-98773892026-04-22T01:39:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-cubic-polynomial--hx--x3--px--q--and-values-5305522026-04-22T01:39:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-equation-76968332026-04-22T01:39:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--2x---b--y--so-the-equation-becomes-44122842026-04-22T01:39:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angles-are-30circ-60circ-and-90circ-using-the-law-of-sines-frac5sin-30circ--fraccsin-90circ-since-sin-30circ--05-and-sin-90circ--1-c--frac505--10-km-73197412026-04-22T01:44:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-circumference-c--2pi-r-so-r--fracc2pi--frac12pi2pi--6-nm-59047672026-04-22T01:44:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-longest-chord-in-a-circle-is-the-diameter-however-the-question-specifies-the-longest-chord-that-is-not-a-diameter-which-would-approach-the-diameter-for-a-circle-the-maximum-chord-length-is-the-diameter-so-the-answer-is-14-cm-47536452026-04-22T01:44:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-hypotenuse-is-sqrt92--122--15-cm-sin-theta--fractextoppositetexthypotenuse--frac915--frac35-70918902026-04-22T01:44:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-largest-number-of-identical-grants-compute-gcd2025-1515-64885992026-04-22T01:44:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-number-of-distinct-sequences-is-the-least-common-multiple-lcm-of-13-and-17-67486892026-04-22T01:44:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-alignment-occurs-at-the-least-common-multiple-of-2025-and-1515-17693062026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x2-equiv-25-pmod100-13671892026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-requires-calculating-the-number-of-distinct-permutations-of-10-items-where-there-are-5-identical-bees-3-identical-wasps-and-2-identical-beetles-the-formula-for-multinomial-coefficients-applies-86414932026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-permutation-problem-the-total-sequences-are-given-by-74426492026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-ways-to-choose-3-species-binom203--1140-the-number-of-favorable-outcomes-is-binom102-cdot-binom61--45-cdot-6--270-the-probability-is-12221502026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-each-discipline-as-a-block-there-are-3-ways-to-arrange-the-blocks-around-the-table-within-the-biology-block-3-physics-2-and-chemistry-2-for-circular-permutations-fix-one-block-to-eliminate-rotational-symmetry-555102026-04-22T01:44:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-ways-to-choose-4-species-binom134--715-favorable-outcomes-binom82-cdot-binom52--28-cdot-10--280-the-probability-is-72609752026-04-22T01:44:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-two-consecutive-sessions-as-a-single-entity-there-are-4-ways-to-arrange-the-4-entities-the-pair-and-the-other-3-sessions-the-pair-can-be-ordered-in-2-ways-thus-total-arrangements-are-4-times-2--24-times-2--48-boxed48-30680812026-04-22T01:54:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-combinations-without-restrictions-dbinom94--126-subtract-combinations-where-both-problematic-startups-are-included-dbinom72--21-choosing-2-more-from-the-remaining-7-valid-combinations-126---21--105-boxed105-78853122026-04-22T01:54:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-inclusion-exclusion-total-assignments-36--729-subtract-assignments-missing-at-least-one-category-3-times-26--192-add-back-assignments-missing-two-categories-3-times-16--3-valid-assignments-729---192--3--540-boxed540question-an-electrical-engineer-tests-a-new-circuit-with-5-identical-resistors-each-with-a-10-probability-of-failing-independently-what-is-the-probability-that-exactly-2-resistors-fail-25975322026-04-22T01:54:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-is-calculated-using-the-binomial-distribution-formula-96791522026-04-22T01:54:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-complementary-probability-of-no-scrolls-containing-the-symbol-is-0958-thus-28687562026-04-22T01:54:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-binomial-formula-6595422026-04-22T01:54:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-5-terms-are--a1--5---a2--8---a3--11---a4--14--and--a5--17--their-sum-is--5--8--11--14--17--55--dividing-55-by-7-gives-a-quotient-of-7-and-a-remainder-of--55---7-times-7--55---49--6--63737962026-04-22T01:54:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x-equiv-1-pmod9--and--x-equiv-1-pmod11--this-implies--x-equiv-1-pmod99--since-9-and-11-are-coprime-the-two-digit-numbers-satisfying-this-are--99--1--100--not-two-digit-and--1--however-the-next-smaller-solution-is--1--which-is-not-two-digit-thus-no-two-digit-number-satisfies-both-conditions-16823792026-04-22T01:54:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-4-primes-after-20-are-23-29-31-and-37-their-sum-is--23--29--31--37--120--dividing-120-by-5-gives-a-remainder-of--120---5-times-24--0--57629422026-04-22T01:54:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-area-of-a-triangle-with-known-side-lengths--a--7---b--10--and--c--13--we-use-herons-formula-74113252026-04-22T01:54:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--a--b--c--1--with--a-b-c--0--so--1---a--b--c---1---b--a--c--and--1---c--a--b--the-expression-becomes-2427402026-04-22T02:36:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--fx--ax2--bx--c--use-the-given-conditions-73073652026-04-22T02:37:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--u--sqrtv--so--v--u2--and--u-geq-0--substitute-31983202026-04-22T02:37:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-slope-of-the-line-from-2-3-to-5-11-is-frac11---35---2--frac83-since-tan-theta-equals-the-slope-tan-theta--frac83-boxeddfrac83-39140752026-04-22T02:37:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-2z--60circ-120circ-420circ-480circ-within-0circ-720circ-yielding--47534862026-04-22T02:37:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-positive-integer-divisible-by-both-7-and-5-we-compute-the-least-common-multiple-lcm-of-7-and-5-since-7-and-5-are-coprime-their-lcm-is-simply-their-product-10285012026-04-22T02:43:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-the-total-number-of-regions-is-12-and-the-number-of-clusters-must-be-an-integer-between-2-and-5-inclusive-we-want-to-maximize-the-number-of-regions-per-cluster-which-means-minimizing-the-number-of-clusters-while-satisfying-the-constraints-69711462026-04-22T02:43:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-180-dots-and-must-arrange-them-into-rows-with-the-same-number-of-dots-where-the-number-of-rows-is-a-prime-number-greater-than-3-to-maximize-the-number-of-dots-per-row-we-must-minimize-the-number-of-rows-among-valid-prime-numbers-22117502026-04-22T02:43:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-divide-both-48-and-72-regions-into-equal-sized-groups-with-no-regions-left-out-we-must-find-the-greatest-common-divisor-gcd-of-48-and-72-28955432026-04-22T02:43:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-number-of-samples-per-column-such-that-the-number-of-columns-is-a-multiple-of-both-6-and-10-we-compute-the-least-common-multiple-lcm-of-6-and-10-89430352026-04-22T02:43:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sine-of--theta--equals--fracsqrt32--at--60circ--and--120circ--in-the-interval--0circ-360circ--these-angles-correspond-to-the-reference-angle--60circ--in-the-first-and-second-quadrants-thus-the-solutions-are--boxed60circ--and--boxed120circ--37829852026-04-22T03:02:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero--2x---13--0--solving--2x---3--0--gives--x--frac32--the-ordered-pair-is--boxedleft-frac32-3-right--26019892026-04-22T03:02:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-cosine-function-oscillates-between---1--and--1--so--5cosleftfracpi6xright--ranges-from---5--to--5--adding-10-shifts-the-range-to--5-15--the-range-is--boxed5-15--51346902026-02-24T20:42:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--theta--arccosleftfrac1sqrt2right--so--cos-theta--frac1sqrt2--then--theta--45circ--and--tan-theta--1--the-answer-is--boxed1--60079742026-04-22T03:02:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression-under-the-square-root-must-satisfy--4---x2-geq-0--so--x-in--2-2--the-maximum-value-of--sqrt4---x2--is--2--at--x--0--and-the-minimum-is--0--at--x--pm-2--the-range-is--boxed0-2-think-68772922026-02-24T20:42:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-for-mathbfv-we-use-the-cross-product-equation-mathbfv-times-mathbfw--mathbfp-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-calculated-as-beginpmatrix-v2-cdot-3---v3-cdot--1--v3-cdot-2---v1-cdot-3--v1-cdot--1---v2-cdot-2-endpmatrix--beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix-setting-this-equal-to-mathbfp--beginpmatrix-5--0---2-endpmatrix-we-get-the-system-42481092026-02-24T20:42:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-mathbfv--beginpmatrix-v1--v2--v3-endpmatrix-the-cross-product-mathbfv-times-mathbfw-is-beginpmatrix-3v2--v3--2v3---3v1---v1---2v2-endpmatrix--beginpmatrix-3--0---2-endpmatrix-this-gives-the-system-44430822026-04-22T03:02:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-12-and-18-factorize-both-12--22-cdot-3-and-18--2-cdot-32-the-lcm-takes-the-highest-powers-of-all-primes-22-cdot-32--4-cdot-9--36-thus-the-lcm-is-boxed36-90302712026-04-22T03:08:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-minimum-depth-where-both-measurements-coincide-is-the-lcm-of-5-and-7-since-5-and-7-are-prime-lcm5-7--5-cdot-7--35-the-answer-is-boxed35-88687402026-04-22T03:08:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-by-summing-the-values-and-dividing-by-the-count-frac10--14--163--frac403-approx-13overline3-the-exact-average-is-boxeddfrac403-2806792026-04-22T03:08:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-expressions-4y---3--2y--5--y--7--7y--9-divide-by-3-and-set-equal-to-10-frac7y--93--10-multiply-both-sides-by-3-7y--9--30-subtract-9-7y--21-divide-by-7-y--boxed3-38585902026-04-22T03:08:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-cosine-of-45circ-is-given-by-cos-45circ--fracsqrt22-for-a-square-with-side-length-s--7-units-the-length-of-the-diagonal-d-can-be-calculated-using-the-pythagorean-theorem-d--ssqrt2-substituting-the-value-of-s-we-get-d--7sqrt2-therefore-the-length-of-the-diagonal-is-boxed7sqrt2-53322122026-04-22T03:10:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-a-of-a-regular-hexagon-with-side-length-s-is-given-by-the-formula-a--frac3sqrt32s2-setting-this-equal-to-54sqrt3-we-have-66933692026-04-22T03:10:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-a-of-the-triangle-is-given-by-79549812026-04-22T03:10:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-v-of-a-cone-is-given-by-50150612026-04-22T03:10:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-an-equilateral-triangle-with-side-length-s-the-radius-r-of-the-circumscribed-circle-is-given-by-1762072026-04-22T03:10:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--sqrtx--y--so--x--y2--substituting-into-the-equation-gives-11043562026-04-22T03:10:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-analyze-the-inequality--x--y--x---y-leq-8--this-expression-is-symmetric-and-represents-a-diamond-rhombus-centered-at-the-origin-96271832026-04-22T03:10:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--fx--ax3--bx2--cx--d--use-the-given-values-to-form-a-system-of-equations-78467352026-04-22T03:10:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--sqrtx--y--so--x--y2--substituting-gives-19658142026-04-22T03:11:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-one-marker-is-mandatory-choose-1-additional-marker-from-the-remaining-3-dbinom31-then-select-2-sequences-from-6-dbinom62-multiply-these-dbinom62-times-dbinom31--15-times-3--45-the-total-number-of-valid-combinations-is-boxed45question-a-tech-startup-founder-has-8-potential-investment-opportunities-and-wants-to-diversify-by-selecting-5-but-two-specific-opportunities-are-mutually-exclusive-how-many-valid-investment-portfolios-can-be-formed-22404342026-04-22T03:20:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-surjective-function-problem-count-the-number-of-ways-to-partition-7-distinct-species-into-4-non-empty-habitats-using-the-inclusion-exclusion-principle--sumk04--1k-binom4k-4---k7--calculating-47---binom4137--binom4227---binom4317-compute-each-term-16384---4-times-2187--6-times-128---4-times-1--16384---8748--768---4--8400-the-final-answer-is-boxed8400question-26507452026-02-24T21:00:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-tan60circ--fractextheighttextshadow-length-we-substitute-tan60circ--sqrt3-and-shadow-length-5textcm-solving-for-height-textheight--5-cdot-sqrt3-thus-boxed5sqrt3-85718832026-04-22T03:20:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-an-equilateral-triangle-is-fracsqrt34-s2-solving-for-original-side-s-fracsqrt34-s2--25sqrt3-rightarrow-s2--100-rightarrow-s--10textcm-new-side-14textcm-new-area-fracsqrt34-cdot-142--49sqrt3-increase-49sqrt3---25sqrt3--24sqrt3-thus-boxed24sqrt3-59685022026-04-22T03:20:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--h--represent-the-number-of-hours-worked-the-total-fee-for-the-first-consultant-is--300--75h--and-for-the-second-consultant-it-is--200--90h--setting-these-equal-17544672026-04-22T03:27:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--dt--kt2--5t--and-that--d3--48--substituting--t--3--16141882026-04-22T03:28:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-altitude-is-modeled-by-the-quadratic-function--y--3x2---12x--15--since-the-coefficient-of--x2--is-positive-the-parabola-opens-upward-and-the-minimum-value-occurs-at-the-vertex-the-x-coordinate-of-the-vertex-is-4853222026-04-22T03:28:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--e--represent-the-number-of-employees-the-total-cost-for-plan-a-is--100--10e--and-for-plan-b-it-is--150--7e--setting-these-equal-88221992026-04-22T03:28:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--a--2b--substitute-into-the-expression-for--e--99543152026-04-22T03:28:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-function--ht---5t2--30t--10--is-a-quadratic-with-a-negative-leading-coefficient-so-the-maximum-occurs-at-the-vertex-the-time--t--at-which-the-maximum-altitude-occurs-is-44901012026-04-22T03:28:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-field-is-120-meters-wide-short-side-and-160-meters-long-long-side-to-ensure-full-coverage-the-drone-flies-parallel-passes-along-the-120-meter-width-with-each-pass-covering-20-meters-in-the-160-meter-direction-the-number-of-passes-required-is-frac12020--6-passes-each-pass-spans-160-meters-in-length-since-the-drone-turns-at-the-end-of-each-pass-and-flies-back-along-the-return-path-each-pass-contributes-160--160--320-meters-of-travelexcept-possibly-the-last-one-if-it-doesnt-need-to-return-but-since-every-pass-must-be-fully-flown-and-aligned-the-drone-must-complete-all-6-forward-and-6-reverse-segments-however-the-problem-states-it-aligns-passes-to-scan-fully-implying-the-drone-flies-each-pass-and-returns-so-6-forward-and-6-backward-segments-but-optimally-the-return-can-be-integrated-into-flight-planning-however-since-no-overlap-or-efficiency-gain-is-mentioned-assume-each-pass-is-a-continuous-straight-flight-and-the-return-is-part-of-the-route-but-standard-interpretation-for-full-coverage-with-back-and-forth-there-are-6-forward-passes-and-5-returns-noproblem-says-to-fully-scan-with-aligned-parallel-passes-suggesting-each-pass-is-flown-once-in-20m-width-and-the-drone-flies-each-160m-segment-and-the-turn-around-is-inherent-but-to-minimize-total-distance-assume-the-drone-flies-each-160m-segment-once-in-each-direction-per-pass-that-would-be-inefficient-but-in-precision-agriculture-standard-for-120m-width-6-passes-at-20m-width-the-drone-flies-6-successive-160m-lines-and-at-the-end-turns-and-flies-back-along-the-return-pathtypically-the-return-is-not-part-of-the-scan-but-the-drone-must-complete-the-loop-however-in-such-problems-its-standard-to-assume-each-parallel-pass-is-flown-once-in-each-direction-unlikely-better-interpretation-the-drone-flies-6-passes-of-160m-each-aligned-with-the-120m-width-and-the-return-from-the-far-end-is-not-counted-as-flight-since-its-typical-in-grid-scanning-but-problem-says-shortest-total-distance-so-we-assume-the-drone-must-make-6-forward-passes-and-must-return-to-start-for-safety-or-data-sync-so-6-forward-and-6-return-segments-each-160m-so-total-distance-6-times-160-times-2--1920-meters-but-is-the-return-160m-yes-if-flying-parallel-but-after-each-pass-it-returns-along-a-straight-line-parallel-so-160m-so-total-6-times-160-times-2--1920-but-waitcould-it-fly-return-at-angles-no-efficient-is-straight-back-but-another-optimization-after-finishing-a-pass-it-doesnt-need-to-turn-180--it-can-resume-along-the-adjacent-160m-segment-no-because-each-160m-segment-is-a-new-parallel-line-aligned-perpendicular-to-the-width-so-after-flying-north-on-the-first-pass-it-turns-west-180-to-fly-south-return-but-thats-still-160m-so-each-full-cycle-pass--return-is-320m-but-6-passes-require-6-returns-only-if-each-turn-around-is-a-complete-180-and-160m-straight-line-but-after-the-last-pass-it-may-not-need-to-returnit-finishes-but-problem-says-to-fully-scan-the-field-and-aligned-parallel-passes-so-likely-it-plans-all-6-passes-each-160m-and-must-complete-them-but-does-it-imply-a-return-the-problem-doesnt-specify-a-landing-or-reset-so-perhaps-the-drone-only-flies-the-6-passes-each-160m-and-the-return-flight-is-avoided-since-its-already-at-the-far-end-but-to-be-safe-assume-the-drone-must-complete-the-scanning-path-with-back-and-forth-turns-between-passes-so-6-upward-passes-160m-each-and-5-downward-returns-160m-each-totaling-6-times-160--5-times-160--11-times-160--1760-meters-but-standard-in-robotics-for-grid-coverage-total-distance-is-number-of-passes-times-width-times-2-forward-and-backward-but-only-if-returning-to-start-however-in-most-such-problems-unless-stated-otherwise-the-return-is-not-counted-beyond-the-scanning-legs-but-here-it-says-shortest-total-distance-so-efficiency-matters-but-no-turn-cost-given-so-assume-only-flight-distance-matters-and-the-drone-flies-each-160m-segment-once-per-pass-and-the-turn-between-is-instantso-total-flight-is-the-sum-of-the-6-passes-and-6-returns-only-if-full-loop-but-that-would-be-12-segments-of-160m-noeach-pass-is-160m-and-there-are-6-passes-and-between-each-a-return-that-would-be-6-passes-and-11-returns-no-clarify-the-drone-starts-flies-160m-for-pass-1-east-then-turns-west-180-flies-160m-return-back-then-turns-north-90-flies-160m-pass-2-etc-but-each-return-is-not-along-the-next-passeach-new-pass-is-a-new-160m-segment-in-a-perpendicular-direction-but-after-pass-1-east-to-fly-pass-2-north-it-must-turn-90-left-but-the-flight-path-is-now-160m-northso-its-a-corner-the-total-path-consists-of-6-segments-of-160m-each-in-consecutive-perpendicular-directions-forming-a-spiral-like-outer-loop-but-actually-orthogonal-the-path-is-160m-east-160m-north-160m-west-160m-south-etc-forming-a-rectangular-path-with-6-sides-no6-parallel-lines-alternating-directions-but-each-line-is-160m-and-there-are-6-such-lines-3-pairs-of-opposite-directions-the-return-between-lines-is-instantaneous-in-2dso-only-the-6-flight-segments-of-160m-matter-but-thats-not-realistic-in-reality-moving-from-the-end-of-a-160m-east-flight-to-a-160m-north-flight-requires-a-90-turn-but-the-distance-flown-is-still-the-160m-of-each-leg-so-total-flight-distance-is-6-times-160--960-meters-for-forward-plus-no-returnsince-after-each-pass-it-flies-the-next-pass-directly-but-to-position-for-the-next-pass-it-turns-but-that-turn-doesnt-add-distance-so-total-directed-flight-is-6-passes--160m--960m-but-is-that-sufficient-the-problem-says-to-fully-scan-so-each-120m-wide-strip-must-be-covered-and-with-6-passes-of-20m-width-its-done-and-aligned-with-shorter-side-so-minimal-path-is-6--160--960-meters-but-waitafter-the-first-pass-east-it-is-at-the-far-west-of-the-120m-strip-then-flies-north-for-160mthis-covers-the-north-end-of-the-strip-then-to-fly-south-to-restart-westward-it-turns-and-flies-160m-south-return-covering-the-south-end-then-east-etc-so-yes-each-160m-segment-aligns-with-a-new-120m-wide-parallel-and-the-160m-length-covers-the-entire-160m-span-of-that-direction-so-total-scanned-distance-is-6-times-160--960-meters-but-is-there-a-return-the-problem-doesnt-say-the-drone-must-return-to-startjust-to-fully-scan-so-960-meters-might-suffice-but-typically-in-such-drone-coverage-a-full-scan-requires-returning-to-begin-the-next-strip-but-here-no-indication-moreover-6-passes-of-160m-each-aligned-with-120m-width-fully-cover-the-area-so-total-flight-6-times-160--960-meters-but-earlier-thought-with-returns-was-incorrectno-separate-returnline-the-flight-is-continuous-with-turns-so-total-distance-is-960-meters-but-lets-confirm-dimensions-field-120m-w--160m-n-each-pass-160m-n-or-s-covering-a-120m-wide-band-6-passes-every-20m-covers-0120m-w-each-at-20m-intervals-020-2040--100120-each-pass-covers-one-120m-wide-strip-the-length-of-each-pass-is-160m-the-length-of-the-field-so-yes-6--160--960m-but-is-there-overlap-in-dense-grid-usually-offset-but-here-no-mention-of-offset-so-possibly-overlapping-but-for-minimum-distance-we-assume-no-redundancyoptimize-path-but-the-problem-doesnt-say-it-can-skip-turnsso-we-assume-the-optimal-path-is-6-straight-segments-of-160m-each-in-a-new-26170932026-02-24T21:08:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-dominant-to-recessive-traits-in-species-a-is--53--this-means-that-for-every-5-dominant-traits-there-are-3-recessive-traits-5180002026-04-22T03:33:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-algae-doubles-in-area-every-3-days-starting-from--50834882026-04-22T03:33:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-red-to-yellow-flowers-is--47--55996852026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-dna-segment-repeats-every-12-base-pairs-and-mutations-occur-every-48-base-pairs-73972892026-04-22T03:33:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-active-compound-to-dried-herb-is--92934492026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-annual-rainfall-is-60-inches-60075402026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio--31--means-3-out-of-every-4-plants-are-tall-and-1-out-of-every-4-is-short-57333992026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-distances-be--a---2d-a---d-a-a--d-a--2d--50186462026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-the-formula--an--3n--k--95860522026-04-22T03:33:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-fossils-found-at-the-five-sites-be-a---2d-a---d-a-a--d-and-a--2d-where-a-is-the-middle-term-and-d-is-the-common-difference-of-the-arithmetic-sequence-the-sum-of-these-terms-is-52434282026-04-22T03:44:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-original-number-of-bees-be-7x-and-butterflies-be-4x-after-adding-18-bees-the-new-number-of-bees-is-7x--18-the-new-ratio-is-given-as-53-so-21457402026-04-22T03:44:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-note-the-only-even-prime-is--2--all-other-primes-are-odd-the-sum-of-three-numbers-is-even-if-57440492026-04-22T03:55:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-four-digit-number-divisible-by-both-7-and-13-we-first-compute-their-least-common-multiple-since-7-and-13-are-both-prime-their-lcm-is-simply-their-product-39172612026-04-22T04:08:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-simplify-reduce-each-base-modulo-8-19685582026-04-22T04:08:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-integers-be--n-n1-n2-n3--their-product-is-51190382026-04-22T04:08:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-point--d--such-that--abcd--is-a-parallelogram-we-use-the-property-that-in-a-parallelogram-the-diagonals-bisect-each-other-therefore-the-midpoint-of-diagonal--ac--must-be-the-same-as-the-midpoint-of-diagonal--bd--1199652026-04-22T04:15:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-volume-of-nutrient-solution-required-multiply-the-amount-needed-per-culture-by-the-number-of-cultures-12476292026-04-22T04:16:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-decrease-in-thickness-is-43740672026-04-22T04:16:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-of-increments-divide-the-total-distance-by-the-size-of-each-increment-54721372026-04-22T04:16:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-population-doubles-every-3-hours-in-12-hours-the-number-of-doubling-periods-is-73076992026-04-22T04:16:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-daily-flow-rate-by-the-number-of-days-46677402026-04-22T04:16:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-outputs-of-the-panels-be--a---12-a---6-a-a--6-a--12--forming-an-arithmetic-sequence-centered-at-the-third-term--a--the-sum-is-99734582026-04-22T04:20:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-ages-be--a---2d-a---d-a-a--d-a--2d--the-youngest-is--a---2d--1200--and-the-oldest-is--a--2d--2000--adding-these-7764022026-04-22T04:20:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-4-samples-42329282026-04-22T04:20:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-counting-the-number-of-onto-functions-from-7-distinct-kits-to-3-students-using-inclusion-exclusion-74954122026-04-22T04:33:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-the-stirling-number-of-the-second-kind-s6-4-which-counts-the-number-of-ways-to-partition-6-distinct-objects-into-4-non-empty-identical-subsets-using-the-recurrence-relation-63514112026-04-22T04:33:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-sum-the-three-values-and-divide-by-3-42330002026-04-22T04:47:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-square-is--144--and-each-rectangle-covers--24--39844572026-04-22T04:47:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-find-the-least-common-multiple-lcm-of-28-and-45-14117442026-04-22T04:51:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-problem-of-distributing-indistinguishable-items-experiments-into-distinguishable-bins-languages-which-is-solved-using-the-stars-and-bars-method-we-want-to-find-the-number-of-non-negative-integer-solutions-to-the-equation-77360042026-04-22T04:52:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-find-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-tokens-is-8-consisting-of-3-red-3-blue-and-2-green-tokens-since-tokens-of-the-same-color-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-35323712026-04-22T04:52:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-12-distinguishable-zones-into-four-categories-with-fixed-counts-4-structural-5-tool-related-2-residential-and-1-none-since-the-zones-are-distinguishable-but-the-artifacts-within-each-category-are-indistinguishable-we-compute-the-number-of-ways-to-assign-the-categories-to-zones-4194742026-04-22T04:52:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-permutations-of-a-multiset-with-9-elements-4-successes-s-3-failures-f-and-2-inconclusive-i-the-number-of-such-distinct-sequences-is-given-by-46715612026-04-22T04:52:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-display-is-circular-and-two-specific-manuscripts-must-be-adjacent-treat-them-as-a-single-unit-in-a-circular-arrangement--n--distinct-objects-can-be-arranged-in--n-1--ways-so-with-6-units-the-pair--5-other-manuscripts-we-have--6-1--5--arrangements-however-the-two-manuscripts-within-the-pair-can-be-ordered-in--2--ways-thus-total-arrangements-are-76342142026-04-22T04:56:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-an-equilateral-triangle-with-side--s--is-83268612026-04-22T04:57:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangular-array-is-inscribed-in-a-circle-of-radius-10-km-so-the-diagonal-of-the-rectangle-equals-the-diameter-of-the-circle-84012682026-04-22T04:57:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-volume-of-a-sphere-of-radius--r--is-18577242026-04-22T04:57:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-centroid-of-a-regular-tetrahedron-is-the-average-of-its-vertices-let-the-fourth-vertex-be-x-y-z-the-centroid-is-55339722026-04-22T04:57:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-integer-be--x--from-the-problem-we-have-35001402026-04-22T04:57:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-drone-starts-at-a-fixed-station-and-returns-to-it-after-visiting-all-others-exactly-once-we-are-counting-circular-permutations-of-6-stations-fixing-one-station-eliminates-rotational-symmetry-so-the-number-of-distinct-paths-is-6---1--5--120-76849252026-04-22T06:42:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-tout-dabord-nous-listons-tous-les-facteurs-de-60-la-factorisation-en-nombres-premiers-de-60-est--94893292026-03-12T06:44:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-tout-dabord-trouvez-la-factorisation-en-nombres-premiers-de-84--77265442026-03-12T06:44:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-pour-trouver-la-moyenne-additionnez-les-longueurs-et-divisez-par-2--65145742026-03-12T06:44:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-of-equations-97192982026-04-24T12:48:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with--a1--1--compute-each-term-step-by-step-48256102026-04-24T12:48:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--y--3x---10--into-the-first-equation-72697632026-04-24T12:49:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--t--intercept-occurs-when--e--0--substitute--e--0--into-the-equation-99976162026-04-24T12:49:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--b--a--3--into-the-equation-18240392026-04-24T12:49:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-the-cubes-of-the-first--n--positive-integers-is-given-by-the-formula-56775272026-04-24T12:49:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-of-equations-from-a---b--1-express-a--b--1-substitute-into-3a--2b--18-10753572026-03-12T08:31:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute-x--3-into-both-functions-32213652026-03-12T08:32:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute-u--4-and-p--10-into-the-equation-18538192026-04-24T12:50:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-consecutive-odd-integers-be--x---2---x--and--x--2--their-sum-is-81541172026-03-12T08:32:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-simplifies-to-23636232026-03-12T08:32:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--d--be-the-number-of-deliveries-the-net-profit-per-delivery-is-91972072026-03-12T08:32:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-square-is-67939322026-03-12T08:32:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a1--3--common-difference--d--6--and-last-term--an--63--using-the-formula--an--a1--n-1d--79152502026-04-24T12:51:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-reduces-to-permutations-of-a-multiset-there-are--7--5--12--total-shards-with-7-identical-peruvian-and-5-identical-mesoamerican-the-number-of-distinct-arrangements-is--45003892026-04-24T12:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-is-equivalent-to-arranging-4-identical-tablets-and-3-identical-vessels-the-number-of-distinct-arrangements-is--86310402026-04-24T12:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-center-rewrite-the-equation-by-completing-the-square-group--x--terms-and--y--terms-43912892026-04-24T12:52:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-two-equations-equal-to-find-the-intersection-32627452026-04-24T12:52:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-functional-equation-31559222026-04-24T12:52:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-triangle-with-sides-a--13-b--14-and-c--15-the-circumradius-r-of-a-triangle-with-sides-a-b-c-is-given-by-the-formula-50453162026-04-22T07:03:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-original-equilateral-triangle-be-s-the-area-a-of-an-equilateral-triangle-is-62696662026-04-22T07:03:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-volume-of-a-sphere-with-radius-2r-is-53053632026-04-22T07:03:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-since-area-is-fixed-and-altitude-is-inversely-proportional-to-base-the-longest-side-is-21-km-so-we-compute-the-altitude-to-this-side-39288052026-04-22T07:03:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-anthropologist-can-only-compare-ritual-pairs-that-originate-from-neighboring-tribes-there-are-exactly-3-such-neighboring-pairs-since-each-comparison-is-between-a-unique-pair-of-rituals-and-no-pair-is-repeated-or-overlap-implied-the-total-number-of-valid-comparisons-is-simply-the-number-of-given-neighboring-pairs-43939112026-04-22T07:03:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-choosing-4-species-from-9-of-which-4-are-rainforest-dominants-and-5-are-not-we-want-exactly-2-rainforest-dominants-and-2-non-rainforest-dominants-the-number-of-favorable-outcomes-is-54701222026-04-22T07:03:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-in-a-triangle-with-side-lengths--a--13---b--14--and--c--15--we-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--is-20201832026-04-24T12:56:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangle-with-dimensions-5-cm-by-12-cm-is-inscribed-in-a-circle-so-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-61812852026-04-24T12:56:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs--a---b--hypotenuse--z--25--and-inradius--c--5--we-use-the-formula-for-the-inradius-of-a-right-triangle-5752192026-04-24T12:56:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-an-equilateral-triangle-with-side--s--is-91466572026-04-24T12:56:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-base-and-each-leg-have-length--x--then-the-perimeter-is--x--x--x--3x--36--so--x--12--45223472026-04-24T12:56:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-determine-the-prime-factorization-of-4800-begin-by-factoring-step-by-step-44584412026-03-12T12:20:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--v--6k--where--k--is-a-positive-integer-then-40915112026-03-12T12:20:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-simplify-the-expression-12660342026-03-12T13:09:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-number-of-integers-from-1-to-100-that-are-congruent-to-3-modulo-7-these-numbers-form-an-arithmetic-sequence-53793032026-04-24T12:58:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-be-of-the-form-1000a--10b--7-ie-ends-in-7-we-want-the-smallest-such-number-divisible-by-13-42959702026-04-22T07:20:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-square-be--s--the-diagonal-of-a-square-divides-it-into-two-congruent-right-triangles-and-the-relationship-between-the-side-length-and-the-diagonal-is-given-by-the-pythagorean-theorem-39622202026-04-22T07:20:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-let-the-sides-be--a--13---b--14---c--15--the-semi-perimeter-is-95185832026-04-22T07:20:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-of-the-cylinder-be--r--and-its-height-be--h--2r--the-cylinder-is-inscribed-in-a-sphere-of-radius-10-so-the-diagonal-from-the-center-of-the-sphere-to-the-top-rim-of-the-cylinder-forms-a-right-triangle-with-legs--r--and--r--and-hypotenuse-equal-to-the-radius-of-the-sphere-10-cm-85005952026-04-22T07:20:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-6-digit-binary-numbers-with-exactly-one-11-block-we-analyze-each-possible-position-of-the-11-and-ensure-no-additional-consecutive-1s-the-11-can-occupy-positions-12-23-34-45-or-56-for-each-case-the-remaining-digits-must-avoid-forming-additional-11-blocks-28578592026-04-22T07:24:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-the-number-of-onto-functions-from-5-species-to-3-zones-calculated-using-inclusion-exclusion-18925452026-04-22T07:23:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-shoelace-formula-for-the-area-of-a-triangle-with-vertices--x1-y1---x2-y2---x3-y3--57409782026-04-22T07:24:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-complex-numbers--z1--x1--y1i--and--z2--x2--y2i--are-orthogonal-if-their-dot-product--x1x2--y1y2--0--71659152026-04-22T07:24:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-variance-of-a-dataset-is-defined-as-the-average-of-the-squared-deviations-from-the-mean-given-that-there-are-12-samples-and-the-sum-of-the-squares-of-the-deviations-is-144-we-compute-the-variance-as-follows-46062642026-04-22T07:24:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-function--pd--3d--7--and-we-are-to-compute-the-total-number-of-microplastic-particles-from-depth--d--1--to--d--10--this-requires-summing--pd--over-integer-values-from-1-to-10-81046362026-04-22T07:24:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-first-term-of-the-arithmetic-sequence-be--a--and-the-common-difference-be--d--the-sequence-has-8-terms-so-the-sum--s8--is-66729712026-04-22T07:24:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-the-number-of-distinct-sequences-in-which-the-drone-can-visit-5-fields-2-wheat-w-2-corn-c-and-1-soybean-s-where-fields-of-the-same-crop-are-indistinguishable-this-is-a-permutation-of-a-multiset-71617972026-04-24T13:00:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-nucleotides-4-as-3-cs-and-2-gs-where-nucleotides-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-83415242026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-6-days-to-cultivation-tasks-3-wheat-w-2-corn-c-and-1-soybean-s-where-the-order-matters-but-tractors-are-indistinct-the-number-of-distinct-sequences-is-the-multinomial-coefficient-94948042026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-5-proteins-2-as-2-gs-and-1-f-with-indistinguishable-items-within-each-type-the-number-of-distinct-sequences-is-25292232026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-the-number-of-distinct-permutations-of-a-multiset-the-total-number-of-icons-is-4--3--2--9-with-repetitions-of-each-type-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-7261782026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-8-individuals-with-repetitions-3-chimpanzees-2-gorillas-and-3-orangutans-the-number-of-distinct-arrangements-is-36678962026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-10-items-with-repetitions-5-of-one-type-3-of-another-and-2-of-a-third-the-number-of-distinct-sequences-is-94685302026-04-24T13:00:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-items-with-repetitions-4-males-3-females-2-juveniles-the-number-of-distinct-sequences-is-79072542026-04-22T07:24:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-we-add-the-three-expressions-and-divide-by-3-37848612026-04-24T13:01:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-icosahedron-all-edges-are-of-equal-length-first-compute-the-distance-between-the-given-points-30030922026-04-24T13:01:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-use-combinations-to-count-favorable-outcomes-over-total-outcomes-55337172026-04-24T13:01:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-each-altitude-has-one-sensor-selecting-3-sensors-with-no-two-from-the-same-altitude-simply-means-choosing-any-3-distinct-sensors-which-is-guaranteed-as-all-sensors-are-at-different-altitudes-39622602026-04-24T13:01:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-robot-arms-are-identical-the-problem-reduces-to-choosing-4-distinct-arms-out-of-6-to-place-the-sensors-however-because-the-arms-are-indistinct-the-order-of-placement-does-not-matter-thus-the-number-of-ways-is-the-number-of-combinations-of-6-arms-taken-4-at-a-time-68193972026-04-24T13:01:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-partitioning-7-distinct-calculators-into-4-non-empty-identical-subsets-the-number-of-ways-is-given-by-the-stirling-numbers-of-the-second-kind--s7-4--using-the-formula-or-recurrence-relation-25904722026-04-24T13:01:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-terms-of-the-arithmetic-sequence-be-a---3d-a---d-a--d-and-a--3d-where-a-is-the-middle-term-and-d-is-the-common-difference-38856102026-04-22T07:37:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-hypotenuse-of-a-right-triangle-with-legs-5-cm-and-12-cm-using-the-pythagorean-theorem-55962822026-04-22T07:43:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-data-entries-form-an-arithmetic-sequence-an--1--3n---1-for-n--1-to-100-the-sum-is-s--99286582026-04-22T07:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-recursive-sequence-13542192026-04-22T07:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-arrangements-of-9-symbols-with-3-identical-is--61319712026-04-24T13:07:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-five-prime-numbers-are-2-3-5-7-and-11-since-these-numbers-are-all-prime-their-least-common-multiple-is-simply-their-product-14909992026-04-24T13:07:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-observe-that-all-the-numbers-are-odd-and-consecutive-odd-numbers-we-can-reduce-each-number-modulo-8-76344212026-04-24T13:07:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-population-doubles-every-3-hours-we-can-model-the-growth-using-exponential-growth-53068052026-04-22T07:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-be--l--and-the-width-be--w--the-perimeter-is-given-by-43420512026-04-22T07:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-least-common-multiple-lcm-of-12-and-18-which-represents-the-first-day-both-strains-share-the-mutation-again-18186212026-04-22T07:59:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-the-first-component-to-the-second-is-53-let--x--be-the-amount-of-the-second-component-then-50155162026-04-22T07:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-be--l--and-the-width-be--w--the-area-is-given-by-64367342026-04-22T07:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-an-arithmetic-progression-with-first-term--a--3--and-common-difference--d--4--the-general-term-of-an-arithmetic-sequence-is-given-by-2683972026-04-22T07:59:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-compute-the-sum-of-the-squares-of-the-first--7--odd-numbers-modulo--10--the-first--7--odd-numbers-are--1-3-5-7-9-11-13--their-squares-are-34004802026-04-22T08:04:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert-from-base-eight-to-base-ten-expand-the-number-as-a-sum-of-powers-of-8-15440082026-04-22T08:08:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-sum-of-squares-9145482026-04-22T08:08:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-trigonometric-values-605552026-04-22T08:17:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-thickness-that-divides-all-measurements-we-compute-the-greatest-common-divisor-gcd-of-all-four-thicknesses-360-480-270-and-540-20092202026-04-22T08:20:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-by-summing-the-transactions-and-dividing-by-the-number-of-departments-83256512026-04-22T08:20:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-sequences-align-we-compute-the-least-common-multiple-lcm-of-18-and-24-99152102026-04-22T08:20:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-length-is-calculated-by-57794702026-04-22T08:20:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-involves-partitioning-15-distinct-grants-into-5-groups-of-3-each-where-the-order-of-grants-within-a-university-does-not-matter-the-number-of-ways-is-given-by-the-multinomial-coefficient-93377862026-03-12T22:35:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-combinatorial-problem-with-adjacency-constraints-we-model-the-grid-as-a-4x4-lattice-where-each-cell-is-p-or-i-and-no-two-horizontally-or-vertically-adjacent-cells-are-both-p-this-is-equivalent-to-counting-binary-matrices-with-no-two-adjacent-1s-p--1-i--0-93726052026-03-12T22:34:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-partitioning-6-distinct-compounds-into-3-non-empty-unordered-subsets-phases-the-number-of-ways-is-the-stirling-number-of-the-second-kind--s6-3--multiplied-by-the-number-of-ways-to-assign-these-subsets-to-3-distinguishable-phases-since-phases-are-distinct-eg-phase-1-2-3-86864292026-03-12T22:36:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-10-distinguishable-countries-into-3-categories-with-specified-counts-4-in-highly-automated-3-in-moderately-affected-and-the-remaining-3-in-minimally-influenced-58868302026-04-24T13:19:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-sensors-where-4-are-ai-based-indistinguishable-among-themselves-3-are-traditional-indistinguishable-among-themselves-and-2-are-hybrid-indistinguishable-among-themselves-80367802026-04-24T13:19:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-7-distinguishable-tribes-into-3-categories-with-fixed-counts-3-agrarian-2-nomadic-and-2-maritime-19308842026-04-24T13:19:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-12-distinguishable-job-roles-into-3-categories-with-fixed-counts-5-ai-driven-4-enhanced-by-ai-and-3-unaffected-32232662026-04-24T13:19:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-8-events-where-3-are-strong-3-are-moderate-and-2-are-weak-since-events-of-the-same-type-are-indistinguishable-the-number-of-sequences-is-96733352026-04-24T13:19:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-9-distinguishable-artifacts-into-3-time-periods-with-fixed-counts-4-ancient-3-classical-and-2-modern-10232792026-04-24T13:19:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--m--be-the-midpoint-of--ab--and--n--be-the-midpoint-of--ac--then-17509292026-04-24T13:19:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-partial-fraction-decomposition-28442122026-04-22T08:53:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-difference--fx1---fx--is-linear--fx--must-be-a-quadratic-56477972026-04-22T08:53:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-radius--r--of-the-circumscribed-circle-of-a-triangle-with-sides--a--5---b--6--and--c--7--we-use-the-formula-75650662026-04-22T08:54:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-triangle-can-be-calculated-using-the-base-and-height-58620552026-04-22T08:53:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-square-inscribed-in-a-circle-the-diagonal-of-the-square-is-equal-to-the-diameter-of-the-circle-let--s--be-the-side-length-of-the-square-then-the-diagonal--d--is-75209512026-04-22T08:53:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is--13--cm-first-compute-the-area--k--using-herons-formula-the-semi-perimeter--s--is-28878352026-04-22T08:53:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-30-of-120-calculate-66886262026-04-22T08:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-daily-energy-production-74694982026-04-22T08:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-total-number-by-the-fraction-45606882026-04-22T08:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-population-by-the-fraction-89560542026-04-22T08:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-right-triangle-with-legs--a---b--and-hypotenuse--c--the-area--a--is-60987122026-04-22T08:57:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--n--5--into-the-sequence-formula-22004282026-04-22T09:01:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides-of--x--75518922026-04-22T09:01:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-longest-side-is-15-cm-to-find-the-altitude-to-this-side-first-calculate-the-area-using-herons-formula-the-semi-perimeter-is-13881922026-04-22T09:02:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-14808272026-04-22T09:09:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-with-radius-3x-is-17090462026-04-22T09:09:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-trigonometric-identities-or-standard-values-97510232026-04-22T09:09:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-a-of-an-equilateral-triangle-with-side-length-s-is-10675322026-04-22T09:09:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-1950-to-2050-is-100-years-or-10-decades-24446172026-03-13T04:22:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term-a--2-and-common-difference-d--3-31007582026-03-13T04:22:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-rise-is-given-by-86914082026-03-13T04:21:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-how-many-are-related-to-renewable-energy-60786902026-03-13T04:22:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-a--410-d--2-85753782026-03-13T04:22:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-speed-grows-exponentially-20510472026-03-13T04:22:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-we-add-the-lengths-of-the-two-stems-and-divide-by-2-8186122026-03-13T04:22:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-we-sum-the-data-usage-values-and-divide-by-the-number-of-departments-10582452026-03-13T04:22:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-difference-in-width-subtract-the-smaller-width-from-the-larger-21256032026-04-22T09:18:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-average-sum-the-times-and-divide-by-the-number-of-measurements-18547342026-04-22T09:18:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-be-w-then-the-length-is-2w-the-perimeter-formula-for-a-rectangle-is-83880962026-04-22T09:18:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-height-we-sum-all-the-individual-heights-98534622026-04-22T09:18:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-devices-follows-an-arithmetic-sequence-where-the-first-term-is-13-and-the-common-difference-is-7-the-formula-for-the-nth-term-is-32687172026-04-22T09:18:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-number-of-schools-divide-the-total-kits-by-the-minimum-per-school-89061932026-04-22T09:18:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--y--intercept-of-a-linear-equation--ex--mx--b--is-the-value-of--ex--when--x--0--substituting--x--0--into--ex--35x--12--we-get--e0--350--12--12--71035022026-04-22T09:20:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s--is-given-by-60370832026-04-22T09:20:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-five-numbers-form-an-arithmetic-sequence-we-write-them-as-93107172026-04-22T09:21:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-both-sides-265232026-04-22T09:21:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-add-the-two-values-and-divide-by-2-57690732026-04-22T09:22:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-the-left-side-83556342026-04-22T09:32:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-length-is-found-by-adding-the-lengths-and-dividing-by-2-92729192026-04-22T09:32:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-tetrahedron-all-edges-are-equal-compute-the-distance-between-given-points-34334942026-04-22T09:32:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-smallest-three-digit-number-n-such-that-51765322026-04-22T09:33:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-point-on-the-line--y--13441662026-04-22T09:40:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-given-vertices-be--a100---b010---c001--and-let--dxyz--be-the-fourth-vertex-of-the-regular-tetrahedron-with-integer-coordinates-in-a-regular-tetrahedron-all-edges-are-equal-9239862026-04-24T13:38:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--c--13--the-radius--r--of-the-incircle-of-a-right-triangle-is-given-by-70166842026-04-22T09:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-5-character-strings-using-the-letters-a-b-c-d-4-letters-such-that-each-letter-appears-at-least-once-this-is-a-classic-inclusion-exclusion-problem-46487852026-04-22T09:43:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-a-rectangle-forms-a-right-triangle-with-its-length-and-width-using-the-pythagorean-theorem-the-diagonal-d-is-calculated-as-16274132026-04-22T09:43:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-hexagon-inscribed-in-a-circle-the-radius-equals-the-side-length-of-the-hexagon-this-is-because-the-hexagon-can-be-divided-into-six-equilateral-triangles-thus-the-radius-is-equal-to-the-side-length-30768982026-04-22T09:43:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-center-complete-the-square-for-both--x--and--y--2643672026-04-22T09:43:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-two-vectors-to-be-perpendicular-their-dot-product-must-be-zero-81717162026-04-24T13:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-regular-hexagon-with-side-length--s--is-given-by-49759792026-04-24T13:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-formula-for-the-inradius-57310652026-04-24T13:41:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-a-team-of-4-experts-from-7-mathematicians-5-physicists-and-6-computer-scientists-with-the-restriction-that-at-least-one-expert-must-come-from-each-discipline-59566682026-04-24T13:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-5-distinct-people-around-a-circular-table-where-rotations-are-considered-identical-in-circular-permutations-the-number-of-arrangements-of--n--distinct-people-is--n-1--37106422026-04-24T13:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-forming-a-6-digit-code-with-the-following-constraints-32636972026-04-24T13:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-a-6-member-committee-from-10-scientists-including-3-ai-experts-and-8-engineers-with-58164302026-04-22T09:53:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-lies-along-the-perpendicular-from-the-point-to-the-line-the-slope-of-the-given-line-is----56053422026-04-24T13:46:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-radius-of-the-largest-circle-that-fits-inside-the-triangle-the-incircle-we-use-the-formula-19881212026-04-24T13:46:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-surface-area-of-a-cone-includes-the-lateral-curved-surface-area-and-the-base-area-the-formula-for-total-surface-area-is-8982212026-04-22T10:07:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-of-elevation--heta--can-be-found-using-the-tangent-function-55616652026-04-24T13:46:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-regular-tetrahedron-with-edge-length--a--is-48950482026-04-24T13:46:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-be--w--meters-and-the-length-be--3w--meters-the-perimeter-is-given-by-19580872026-04-24T13:46:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-2000-2004-2008--2024-this-is-an-arithmetic-sequence-with-first-term--a--2000--common-difference--d--4--and-last-term--l--2024--the-number-of-terms--n--is-given-by-23168222026-04-24T13:49:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-remainder-when-the-sum-of-the-patent-numbers-20250414--20250608--20250702--20250796-is-divided-by-100-we-only-need-the-last-two-digits-of-each-number-24507342026-04-22T10:15:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-the-squares-of-the-first-n-positive-integers-is-given-by-the-formula-40057282026-04-22T10:15:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by-10-to-eliminate-denominators-82520822026-04-22T10:21:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--y--intercept-occurs-when--x--0--substitute--x--0--into-the-equation-770132026-04-22T10:21:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-terms-be--a---2d-a---d-a-a--d-a--2d--the-fifth-term-is--a--2d--and-the-first-is--a---2d--their-sum-is-40826262026-04-22T10:24:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-expressions-equal-48589332026-04-22T10:24:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-distributive-property-foil-method-6993332026-04-22T10:28:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-of-the-rectangle-be--w--meters-then-the-length-is--3w--meters-the-perimeter-is-given-by-65714352026-04-24T13:54:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-weights-of-the-two-foxes-be--x--and--y--we-are-given-21122012026-04-22T10:37:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a1--3--and-common-difference--d--5--the--n--th-term-is-given-by--an--a1--n-1d--for--n--7--71347982026-04-22T10:45:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--3--into-the-equation-682112026-04-22T10:45:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-geometric-with--a1--1---r--2--the-sum-of--n--5--terms-is-10091552026-04-22T10:45:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-how-many-4-digit-numbers-are-divisible-by-11-47377202026-04-22T10:49:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-robots-net-position-subtract-the-backward-distance-from-the-forward-distance-71033732026-04-22T10:49:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-add-the-two-lengths-and-divide-by-2-70841502026-04-22T10:49:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-5-digit-number-is--10000--the-largest-is--99999--81179312026-04-22T10:49:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-the-triangle-is-1554432026-04-22T10:54:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-elements-is-6--4--3--13-since-items-within-each-category-are-indistinguishable-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-27227462026-04-22T10:58:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-11-positions-where-5-are-clay-c-2-are-silt-s-and-4-are-loam-l-with-indistinguishability-within-each-type-the-number-of-distinct-sequences-is-the-multinomial-coefficient-7304052026-04-22T10:58:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-there-are-5--3--2--10-days-total-the-number-of-distinct-sequences-is-given-by-77013402026-04-22T10:58:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-devices-is-4--3--3--10-with-indistinguishability-within-each-category-the-number-of-distinct-sequences-is-40495452026-04-22T10:58:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-15-items-9-identical-observations-o-4-identical-thought-experiments-t-and-2-identical-historical-cases-h-the-number-of-distinct-sequences-is-50546422026-04-22T10:58:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-count-the-number-of-valid-lattice-paths-from-0-0-to-5-4-that-do-not-cross-the-line--y--x--1--we-use-the-reflection-principle-first-compute-the-total-number-of-unrestricted-paths-from-0-0-to-5-4-48205112026-04-22T11:00:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-intersection-set-the-two-expressions-for--y--equal-41583262026-04-22T11:02:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--represent-the-number-of-hours-after-which-the-volumes-are-equal-35287822026-04-24T14:03:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-two-numbers-be--a--and--b--we-are-given-85218132026-04-22T11:15:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-two-variables-be--a--and--b--we-are-given-80539572026-04-22T11:15:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-the-growth-rate-is-zero-solve-the-equation-24884912026-04-22T11:15:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initially-42-out-of-60-startups-were-successful-after-10-more-evaluations-the-total-number-of-startups-becomes-10866732026-04-22T11:15:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-square-is--144--and-each-rectangle-has-area--12--dividing-88358042026-04-22T11:18:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-magnitude-of-the-sum-of-vectors-59929352026-04-22T11:18:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-compute-the-remainder-when-the-sum-11081--11083--11085--11087-is-divided-by-8-first-observe-that-these-are-four-consecutive-odd-numbers-we-can-reduce-each-number-modulo-8-to-simplify-63792042026-04-22T11:27:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-tasked-with-finding-the-number-of-unique-test-combinations-consisting-of-3-distinct-viral-strains-chosen-from-7-and-2-antiviral-compounds-chosen-from-8-since-the-choices-are-independent-we-compute-the-product-of-the-combinations-60163442026-04-22T11:27:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-fix-one-scientist-at-a-seat-to-eliminate-rotational-symmetry-in-circular-permutations-this-leaves-4-scientists-to-arrange-in-the-remaining-seats-which-can-be-done-in-4-ways-8339682026-04-22T11:27:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-fourth-vertex-be--d--x-y-z--since-the-tetrahedron-is-regular-all-edges-must-have-equal-length-first-compute-edge-lengths-between-the-given-points-51266372026-03-13T19:20:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x-and-y-be-the-number-of-patients-in-the-first-and-second-groups-then--57134192026-04-22T11:33:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-expressions-and-divide-by-3-60720812026-04-24T14:11:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-inequality-step-by-step-4500812026-04-22T11:38:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-e3--r3-first-compute-e3-53184592026-04-22T11:38:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-v12--v22-compute-v12-10584892026-04-24T14:11:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-parallelogram-the-diagonals-bisect-each-other-therefore-the-midpoint-of-diagonal--ac--must-be-the-same-as-the-midpoint-of-diagonal--bd--93244452026-04-24T14:11:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-analyze-the-function-56524572026-04-24T14:14:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-express-the-function-in-terms-of-double-angle-identities-recall-51946232026-04-24T14:14:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-duration-of-the-signal-as-observed-from-earth-we-use-the-time-dilation-formula-from-special-relativity-65432502026-03-13T23:16:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the--x--intercept-we-set--y--0--and-solve-for--x--first-find-the-slope-of-the-line-through-the-points--2-5--and--6-13--90959482026-03-13T23:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute--f3--and--g3--41234372026-03-13T23:17:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-sequence-is-arithmetic-with-first-term--a--3--and-common-difference--d--5--the-last-term-is--l--198--using-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-58334942026-04-22T12:09:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term--a--2--common-difference--d--5--and-last-term--l--197--using-the--n--th-term-formula-90013342026-04-22T12:10:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product-71011442026-04-22T12:12:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-is-the-perpendicular-projection-of-that-point-onto-the-line-70034712026-04-22T12:12:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-be--x--cm-then-the-length-is--x--4--cm-the-perimeter-is-given-by--2x--x--4--80--which-simplifies-to-78370712026-03-14T00:37:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-width-be--x--inches-then-the-length-is--2x--inches-the-perimeter-is-45300292026-03-14T00:37:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-functional-equation-known-as-cauchys-functional-equation-in-three-variables-we-analyze-it-step-by-step-67530092026-03-14T00:37:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-average-sum-the-two-values-and-divide-by-2-90926342026-04-22T12:17:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a1--5--common-difference--d--4--and-last-term--an--41--54753262026-04-22T12:17:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-equation-for-the-average-of-the-two-expressions-84420512026-04-24T14:23:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-of-migration-events-divide-the-total-days-by-the-cycle-length-84359362026-04-24T14:23:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-least-common-multiple-lcm-of-8-and-12-determines-when-both-events-align-prime-factorizations-28069732026-04-24T14:23:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-lcm-of-6-and-9-is-calculated-as-follows-94845772026-04-24T14:23:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-desired-integer-be--x--3227042026-04-22T12:23:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-blue-markers-be-x-the-ratio-54-implies-66005902026-04-24T14:25:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-mean-is-calculated-by-summing-the-values-and-dividing-by-2-49807102026-04-24T14:25:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-reactions-are-independent-the-probability-that-all-three-succeed-is-35966912026-04-24T14:25:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-n--4-so-the-deck-has-3n--12-cards-labeled-from-1-to-12-we-want-the-probability-that-the-sum-of-3-randomly-drawn-distinct-cards-is-divisible-by-3-14119222026-04-24T14:25:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--1--into-both-functions-71188322026-04-22T12:28:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-equations-equal-at--x--3--88485562026-04-22T12:28:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-coefficient-the-number-of-ways-is--8993372026-04-22T12:29:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-prism-is-given-by-the-base-area--a--multiplied-by-the-height--h--thus--v-80578282026-04-24T14:29:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-is-calculated-as--94057182026-04-24T14:30:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-each-number-modulo-13-25168502026-04-24T14:30:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-team-has-4-prototypes-each-with-a-distinct-rank-from-1-to-4-the-third-team-has-3-prototypes-ranked-1-to-3-since-one-prototype-is-selected-uniformly-at-random-from-each-team-the-selection-within-each-team-is-independent-41214042026-04-22T12:45:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-a-total-of--5--2--1--8--atoms-where-5-are-identical-hydrogen-h-2-are-identical-oxygen-o-and-1-is-nitrogen-n-the-number-of-distinct-permutations-is-given-by-the-multinomial-coefficient-77061622026-04-22T12:45:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-78-and-104-we-start-by-finding-their-prime-factorizations-64267762026-04-24T14:31:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-three-expressions-is-given-by-84874122026-04-24T14:31:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--15--into-the-equation-6728352026-04-22T12:52:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-lengths-be--3x---5x--and--7x--the-sum-is--3x--5x--7x--15x--45--solving-for--x--45075012026-04-22T12:52:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-rectangular-trench-is-inscribed-in-the-circular-foundation-the-diagonal-of-the-rectangle-is-equal-to-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-diagonal-d-is-6736602026-04-22T12:52:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangle-is-inscribed-in-the-circle-so-its-diagonal-is-the-diameter-the-diagonal-d-is-33912092026-04-22T12:52:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-simplify-the-expression--fx--7813532026-04-22T13:03:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-sum-all-the-measurements-and-divide-by-the-number-of-days-82285362026-03-14T06:02:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-mixture--15--35--50-liters-fraction-of-rainwater---60157452026-03-14T06:03:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-there-are-only-two-types-high-income-and-low-income-it-is-impossible-to-select-3-zones-of-different-types-without-including-at-least-two-of-the-same-type-therefore-the-event-of-selecting-3-zones-all-of-different-types-is-impossible-90745372026-03-14T06:03:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-intersection-point-solve-the-system-of-equations-21768932026-03-14T06:03:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-nous-utilisons-la-formule-daddition-pour-la-tangente--42448262026-04-22T13:08:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-distributive-property-to-expand-25653172026-04-22T13:11:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-area-of-a-triangle-with-sides-a--7-b--10-and-c--13-we-use-herons-formula-88854812026-03-14T07:04:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-vectors-to-be-orthogonal-their-dot-product-must-be-zero-compute-3-2--x5--0-this-simplifies-to--6--5x--0-solving-gives-x--74612972026-04-22T13:25:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-chips-is-5--7--3--15-the-probability-the-first-chip-is-amber-is--96237432026-04-22T13:26:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-8-cm-compute-area-using-herons-formula--s--58464292026-04-22T13:26:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-1080-we-have-48731982026-04-24T14:43:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-largest-number-n--200-such-that-49341122026-04-24T14:43:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-using-herons-formula-the-semiperimeter-is-24281812026-04-24T14:45:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given-hypotenuse--25--one-leg--24--the-other-leg-is-76909852026-04-24T14:45:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-involves-distributing-4-distinct-objects-into-2-identical-boxes-this-corresponds-to-the-sum-of-stirling-numbers-of-the-second-kind-for-2-boxes-considering-non-empty-subsets-the-formula-is--s41--s42--here--s41--1--all-groups-in-one-zone-and--s42--7--split-into-two-non-empty-groups-thus-the-total-number-of-ways-is--1--7--8--54093852026-04-22T13:37:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-circular-arrangements-of-n-distinguishable-people-the-number-of-distinct-arrangements-is-n-1-here-there-are-8--5--13-individuals-thus-the-number-of-distinct-seating-arrangements-is-13-1--12-46482972026-04-22T13:38:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinguishable-objects-into-exactly-4-non-empty-indistinguishable-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-s6-4-from-known-values-s6-4--65-since-the-cabinets-are-indistinguishable-we-do-not-multiply-by-permutations-10470192026-04-22T13:38:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-where-the-first-term--a--3--the-common-difference--d--4--and-the-last-term--l--99--to-find-the-number-of-terms--n--we-use-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-56432942026-04-22T13:38:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-to-find-how-many-of-these-are-divisible-by-11-we-find-the-smallest-and-largest-multiples-of-11-in-this-range-26440392026-04-22T13:38:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-count-the-positive-divisors-of-60-the-prime-factorization-of-60-is-33860302026-04-22T13:38:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-center-rewrite-the-equation-by-completing-the-square-group-x-and-y-terms-20056402026-04-22T13:38:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-system-of-congruences-27616332026-03-14T10:18:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-3-toucans-t-2-tanagers-a-1-macaw-m-and-1-hummingbird-h-and-we-want-to-count-the-number-of-distinct-sequences-of-length-7-using-these-indistinguishable-items-the-total-number-of-distinct-permutations-of-a-multiset-is-given-by-20839222026-04-22T13:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-4-identical-robots-r-3-identical-detectors-d-and-2-identical-sensors-s-to-9-distinct-stations-with-one-device-per-station-this-is-a-multinomial-coefficient-problem-87103012026-04-22T13:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-choose-2-different-flavors-from-8-the-number-of-combinations-is-45599442026-04-22T13:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-2-jaguars-j-2-capybaras-c-and-2-sloths-s-all-indistinguishable-within-species-the-number-of-distinct-permutations-is-31878792026-04-22T13:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-5-distinguishable-objects-wires-into-3-indistinguishable-non-empty-boxes-this-is-equivalent-to-finding-the-stirling-numbers-of-the-second-kind-s53-which-counts-the-number-of-ways-to-partition-5-labeled-items-into-3-non-empty-unlabeled-subsets-from-known-values-51513272026-04-22T13:43:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-an-equilateral-triangle-with-side-length-6-km-and-we-are-to-find-the-perpendicular-distance-from-a-vertex-to-the-opposite-sidethis-is-the-altitude-of-the-equilateral-triangle-79461172026-04-22T13:43:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-has-side-lengths-a--13-b--14-c--15-the-longest-side-is-15-cm-so-the-shortest-altitude-corresponds-to-this-side-71789602026-04-22T13:43:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-sphere-is-760672026-03-14T10:53:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-verify-the-triangle-is-right-angled-25742542026-03-14T10:53:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--3427--to-base-ten-we-expand-it-using-powers-of-7-68687832026-04-22T13:53:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-7-and-11-are-coprime-their-least-common-multiple-is-85789512026-04-24T14:50:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-length-is-calculated-by-summing-the-lengths-and-dividing-by-2-94762102026-04-22T14:00:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-cost-of-one-sensor-87345572026-04-22T14:00:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-numerical-values-43375822026-04-22T14:00:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-cost-of-three-smartwatches-multiply-the-cost-of-one-by-3-3071642026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-add-the-three-expressions-and-divide-by-3-78616552026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-red-marbles-be-4x-and-blue-marbles-be-7x-since-4x--28-solve-for-x-87926892026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-amount-each-friend-pays-divide-the-total-cost-by-4-71964152026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-yellow-candies-be-5x-and-orange-candies-be-8x-since-8x--40-solve-for-x-67660272026-04-22T14:08:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-multiset-of-12-soil-cores-3-from-plot-a-2-from-plot-b-and-1-each-from-plots-c-d-e-and-f-the-total-number-of-distinct-sequences-arrangements-in-which-these-indistinct-items-can-be-arranged-is-a-multinomial-coefficient-98866862026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-sequences-of-12-blocks-where-there-are-5-red-r-4-blue-b-and-3-green-g-and-blocks-of-the-same-color-are-indistinguishable-this-is-a-permutation-of-a-multiset-36763652026-04-22T14:08:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-volume-of-a-sphere-with-radius--r--2y--72939502026-04-24T14:55:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-distance-from-a-point-x0-y0-to-a-line-defined-by-ax--by--c--0-is-given-by-the-formula-45117892026-04-22T14:14:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-two-area-functions-67983332026-04-22T14:14:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-first-term-be-a-and-the-common-difference-be-d-82219392026-04-22T14:14:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-ways-to-choose-3-compounds-from-8-and-2-delivery-mechanisms-from-5-we-use-combinations-81572932026-04-22T14:15:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-oak-pollen-grains-be--x--the-ratio--75--implies-78957672026-04-22T14:32:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-white-flowered-plants-be--y--the-ratio--53--implies-63258562026-04-22T14:32:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-dominant-monkeys-be--z--the-ratio--411--implies-45744992026-04-22T14:32:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-an-arithmetic-progression-16239502026-04-22T14:32:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-seating-is-circular-and-the-2-newtonian-experts-must-sit-together-we-treat-them-as-a-single-block-this-reduces-the-problem-to-arranging-6-units-the-block-and-5-historians-in-a-circle-the-number-of-circular-permutations-of-n-objects-is-n-1-so-we-fix-the-block-and-arrange-the-5-historians-around-it-giving-6---1--5-arrangements-within-the-block-the-2-newtonian-experts-can-be-arranged-in-2-ways-thus-the-total-number-of-arrangements-is-10493542026-04-22T14:32:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--c--be-coding-rate-and--t--testing-rate-the-system-is-41578522026-04-24T15:01:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-y-intercept-set--h--0--86951332026-04-24T15:01:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression-simplifies-to-88112942026-04-22T14:42:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given-the-average-detection-over-three-locations-is-20-ppm-we-set-up-the-equation-for-the-average-98836542026-04-22T14:46:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-processing-rate-is-given-by-20527942026-04-22T14:46:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-pressure-change-is-given-by-21450842026-04-22T14:46:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-kale-to-spinach-is-75-meaning-for-every-7-kale-plants-there-are-5-spinach-plants-since-there-are-35-kale-plants-we-determine-how-many-groups-of-7-are-in-35-26342342026-04-24T15:05:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-basil-to-mint-channels-is-43-so-for-every-4-basil-channels-there-are-3-mint-channels-since-there-are-28-basil-channels-we-calculate-the-number-of-groups-of-4-in-28-72304052026-04-24T15:05:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-1660-and-1666-factorize-both-numbers-70655382026-04-24T15:05:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-74-means-7-parts-wheat-correspond-to-4-parts-barley-45546092026-03-14T19:29:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-pour-trouver-la-valeur-minimale-de--15741102026-03-14T19:39:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-path-is-the-difference-between-the-area-of-the-larger-circle-garden--path-and-the-garden-itself-the-radius-of-the-larger-circle-is--r--w--thus-the-area-of-the-path-is-79100262026-04-22T14:52:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-expression-equal-to-5-and-solve-for--x--68361792026-04-22T14:52:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--1--into-the-polynomial-60221052026-04-22T14:52:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--3--into-the-quadratic-equation-99930012026-04-22T14:52:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--6--into-the-function-64032182026-04-22T14:52:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--0--into-the-function-26270292026-04-22T14:52:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--4--into-the-function-85399842026-04-22T14:52:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-toucans-be--t--and-the-number-of-parrots-be--p--from-the-given-ratio-we-have-34496962026-04-22T15:03:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-formula-for-the-area-of-a-triangle-with-coordinates-65487902026-04-22T15:06:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-3-applications-at-random-without-replacement-from-120-total-and-we-want-the-probability-that-each-application-is-from-a-distinct-fieldone-from-cancer-one-from-neuroscience-and-one-from-infectious-diseases-40163462026-04-24T15:10:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-divisor-gcd-of-144-and-60-we-first-perform-the-prime-factorization-of-each-number-93083062026-04-22T15:07:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-cost-of-one-sensor-by-dividing-the-total-cost-by-the-number-of-sensors-42374982026-04-22T15:07:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-5-digit-number-is-10000-and-the-largest-is-99999-80313002026-04-22T15:07:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-area-of-the-square-24921462026-04-22T15:07:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a1--2--common-difference--d--3--and-last-term--an--50--using-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-574922026-03-14T21:08:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-energy-after--n--months-is-the-sum-of-the-arithmetic-series-63044522026-03-14T21:08:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--2--and--y---1--into-the-expression-for--e--18527822026-03-14T21:08:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-recalling-the-identity-91328252026-03-14T21:08:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-integer-we-have-50252432026-04-22T15:16:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-distance-we-add-the-two-mixed-numbers-80880632026-04-22T15:17:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-flow-rate-add-the-two-mixed-numbers-52674562026-04-22T15:17:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-12-and-18-list-their-prime-factorizations-75237272026-04-22T15:17:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-difference-subtract-the-smaller-length-from-the-larger-76180502026-04-22T15:17:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-add-the-three-depths-and-divide-by-3-76589232026-04-22T15:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-prime-factorizations-73729392026-04-22T15:17:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-2400-and-3600-use-prime-factorization-4173702026-04-22T15:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by--2a---1--to-eliminate-the-denominator-89775212026-04-22T15:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-out-the-greatest-common-factor-78865142026-04-22T15:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides-of-the-given-equation-2292192026-04-22T15:17:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-regular-hexagon-be--s--the-distance-from-the-center-to-a-vertex-the-radius-of-the-circumscribed-circle-is-equal-to-the-side-length--s--thus-the-radius--r--of-the-moat-is--s--the-area-of-the-circle-is-38313282026-04-22T15:18:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-right-triangle-have-legs--a---b--and-hypotenuse--h--the-area--a--of-the-triangle-is--31044632026-04-22T15:18:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-geographer-has-a-total-of-9-data-samples-3-satellite-s-2-drone-d-2-ground-g-and-2-aerial-a-since-the-samples-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-88862982026-04-22T15:24:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-student-is-arranging-a-total-of-5--4--3--12-blocks-with-indistinguishability-within-each-color-the-number-of-distinct-towers-is-the-multinomial-coefficient-4597982026-04-22T15:24:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ichthyologist-observes-7-fish-over-7-days2-clownfish-c-3-parrotfish-p-and-2-angelfish-a-since-fish-of-the-same-species-are-indistinguishable-the-number-of-distinct-observation-sequences-is-60235112026-04-22T15:24:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-fix-one-tree-to-account-for-rotational-symmetry-since-the-trees-are-distinguishable-fixing-one-eliminates-rotational-duplicates-the-remaining-4-distinguishable-trees-can-be-arranged-in-4--24-ways-around-the-circle-97550432026-04-22T15:24:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-ways-to-select-5-initiatives-with-at-least-2-high-impact-ones-we-calculate-the-combinations-for-2-and-3-high-impact-initiatives-20468552026-04-22T15:28:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-each-term-in-the-first-binomial-by-each-term-in-the-second-binomial-15457872026-04-22T15:31:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-system-of-equations-75038202026-04-22T15:31:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-of-equations-from-the-second-equation-x--2y---3-substitute-into-the-first-equation-16059452026-04-22T15:31:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-finding-the-number-of-ways-to-partition-6-distinguishable-objects-into-4-non-empty-indistinct-subsets-this-corresponds-to-the-stirling-number-of-the-second-kind--s64--the-value-of--s64--is-calculated-as-follows-20337372026-03-14T23:00:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-combination-problem-where-we-choose-3-startups-from-5-without-regard-to-order-the-number-of-ways-is-given-by-the-binomial-coefficient-97325692026-03-14T23:00:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-word-environment-has-10-letters-with-vowels-e-o-i-e-4-vowels-including-a-repeated-e-and-consonants-n-v-r-n-m-t-6-consonants-with-a-repeated-n-treat-the-vowels-as-a-single-unit-so-we-have-7-units-6-consonants--1-vowel-group-the-number-of-arrangements-is-45923122026-03-14T23:00:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-four-consecutive-odd-integers-are-11071-11073-11075-11077-their-sum-is-59398332026-04-22T15:32:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-least-common-multiple-of-18-and-24-first-factor-both-40638662026-04-22T15:32:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-is-a-right-triangle-with-base-along-the-x-axis-from--a--to--b--and-height-from--a--to--c--47335472026-04-22T15:41:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-circle-enclosing-an-equilateral-triangle-is-its-circumcircle-16213302026-04-24T15:20:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-hypotenuse-of-the-triangle-is-the-diameter-of-the-circumscribed-circle-67846542026-04-24T15:20:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-isosceles-triangle-with-equal-sides-of-10-cm-and-base-12-cm-72678062026-04-22T15:41:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-verify-its-a-right-triangle-91220352026-04-22T15:41:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-remainder-theorem-substitute--v--1--into-the-polynomial-5500672026-04-22T15:55:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-divisor-gcd-of--78--and--130--we-use-the-euclidean-algorithm-49715522026-04-22T15:59:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-where-the-first-term-is--a--3--the-common-difference-is--d--4--and-the-last-term-is--l--99--21518132026-04-22T15:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-the-amount-used-from-the-original-amount-12760912026-04-22T15:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-two-numbers-is-the-sum-divided-by-2-17103202026-04-22T15:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-centroid-g-of-the-triangle-with-vertices-a12-b46-c72-the-centroid-is-76439082026-04-22T15:59:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-horizontal-asymptote-of-the-rational-function--tm--82492082026-04-22T16:02:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-test-divisibility-starting-from-the-smallest-primes-76196072026-04-24T15:26:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a--3--common-difference--d--5--and-last-term--l--198--the-formula-for-the--n-th-term-of-an-arithmetic-sequence-is-94129632026-04-24T15:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-of-number-of-blocks-per-layer-forms-a-sequence-where-the-first-term-is-2-layer-1-and-the-differences-between-terms-increase-by-3-each-time-54246292026-03-15T01:15:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-red-to-blue-pixels-is--74--given-28-red-pixels-set-up-a-proportion-11266462026-04-24T15:26:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with--a--5000---d--500---l--9500--use-69470032026-04-24T15:26:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-starts-at-1-and-the-differences-increase-by-3-5605672026-04-24T15:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio--58--means-for-every-5-males-there-are-8-females-given-40-males-scale-the-ratio-28476452026-04-24T15:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with--a--20---d--4---n--10--use-91686682026-04-24T15:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-out-the-greatest-common-factor-which-is-2-24127392026-04-22T16:03:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term--a--2--common-difference--d--4--and-last-term--l--202--16170712026-04-22T16:03:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate--f-2--30000062026-03-15T01:22:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-distributive-property-to-expand-12241842026-03-15T01:22:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-efficiency-over-25-days-at-72-4917122026-03-15T01:21:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--5728--to-base-ten-expand-using-powers-of-8-82891612026-04-22T16:12:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-semicircle-any-triangle-inscribed-with-the-hypotenuse-as-the-diameter-is-a-right-triangle-since-the-triangle-is-isosceles-and-right-its-legs-are-equal-and-the-hypotenuse-is-the-diameter-24739012026-03-15T02:03:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-sector-is-given-by-14614262026-03-15T02:03:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-distance-between-opposite-vertices-of-a-regular-hexagon-is-twice-the-side-length--s--so-97988622026-03-15T02:03:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-cone-is-8747262026-03-15T02:03:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-smallest-divisor-of-48-greater-than-1-since-each-block-must-contain-more-than-one-state-to-reduce-redundancy-the-divisors-of-48-are-1-2-3-4-6-8-12-16-24-48-the-smallest-divisor-greater-than-1-is-2-2858112026-04-22T16:16:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-find-the-greatest-common-factor-gcf-of-36-54-and-81-prime-factorizations-58100702026-04-22T16:16:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-law-of-cosines-to-find-the-largest-angle-opposite-the-longest-side-13-cm-45797062026-04-22T16:22:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-support-beam-stretches-from-one-point-on-the-edge-of-the-circular-base-through-the-center-to-the-opposite-point-so-its-length-is-the-diameter-39752026-04-22T16:23:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-radius-r-of-the-circumscribed-circle-of-a-triangle-with-sides-a--50-b--60-and-c--70-we-use-the-formula-80613972026-04-22T16:23:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-radius-of-the-base-is-half-the-diameter-18840532026-04-22T16:23:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-triangle-with-sides-7-24-and-25-is-a-right-triangle-because-17870382026-04-22T16:23:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-three-expressions-and-divide-by-3-45005862026-04-22T16:24:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-3-digit-number-divisible-by-7-is-105--7-imes-15-and-the-largest-is-994--7-imes-142-22958152026-04-22T16:24:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-three-expressions-8b---1--5b--4--2b---7--15b---4-34778782026-04-22T16:24:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-concentrations-at-each-hour-are-55438412026-03-15T03:07:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-of-dates-in-bce-is-an-arithmetic-sequence-starting-at-1050-decreasing-by-5-ending-at-1000-91750692026-03-15T03:07:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-we-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-85197782026-03-15T03:07:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-82524642026-03-15T03:07:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is--a-ad-a2d-a3d-a4d--the-middle-term-is--a2d--5--so-13021882026-04-22T16:38:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-apply-the-am-gm-inequality-39978312026-04-22T16:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-difference-gx--1---gx-is-linear-gx-must-be-a-quadratic-polynomial-let-30892012026-04-22T16:42:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-function-is-a-quadratic-with-a-minimum-at--x--28246472026-04-22T16:44:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-3-species-out-of-7-and-4-of-them-are-known-to-use-quantum-entanglement-based-communication-we-want-all-3-selected-species-to-belong-to-this-group-5178172026-04-22T16:58:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangular-sensor-section-is-inscribed-within-the-circular-cross-section-of-the-ice-core-since-the-rectangle-is-inscribed-in-a-circle-its-diagonal-is-equal-to-the-diameter-of-the-circle-applying-the-pythagorean-theorem-to-the-rectangle-with-sides-4-cm-and-8-cm-95394472026-04-22T17:03:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangular-platform-is-inscribed-in-the-circular-foundation-so-the-diagonal-of-the-rectangle-equals-the-diameter-of-the-circle-using-the-pythagorean-theorem-37374972026-04-22T17:03:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diameter-of-the-circular-unit-is-equal-to-the-diagonal-of-the-3-cm-by-4-cm-rectangle-using-the-pythagorean-theorem-52875762026-04-22T17:03:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-m-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-s-is-81173032026-04-22T17:03:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-verify-that-the-triangle-is-right-angled-check-88933332026-04-22T17:03:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-a-be-the-set-of-bees-and-b-the-set-of-butterflies-50092382026-04-22T17:07:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-3-from-10-is-60582342026-04-22T17:07:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-radius--r--of-the-circumscribed-circle-of-a-triangle-with-sides--a--13---b--14--and--c--15--we-use-the-formula-641782026-04-22T17:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-length-of-the-shortest-altitude-corresponds-to-the-longest-side-which-is-11-km-to-find-this-altitude-we-first-compute-the-area-using-herons-formula-the-semi-perimeter--s--is-73039192026-04-22T17:17:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-cubic-equation-86480612026-04-22T17:17:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-lies-along-the-perpendicular-from-the-point-to-the-line-the-slope-of-the-given-25209112026-04-24T15:47:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-samples-is-5--3--2--10-since-samples-of-the-same-type-are-indistinguishable-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-24881032026-04-22T17:34:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-philosopher-has-4--3--7-distinct-items-treating-each-theory-and-framework-as-unique-the-number-of-ways-to-arrange-them-is-the-number-of-permutations-of-7-distinct-objects-32620242026-04-22T17:34:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-of-a-triangle-with-sides-a--13-b--14-and-c--15-we-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-98688182026-03-15T06:51:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-herons-formula-to-compute-the-area-side-lengths-a--10-b--17-c--21-semi-perimeter-82043322026-03-15T06:51:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-center-complete-the-square-for-both--x--and--y--start-by-grouping-terms-17044152026-03-15T06:51:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-is-found-by-minimizing-the-distance-the-distance-d-from-a-point-x1-y1-to-a-point-x-y-on-the-line-y--2x--3-is-41448012026-04-22T17:36:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-point-is-along-the-perpendicular-the-slope-of-the-given-line-is----67752312026-03-15T07:14:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-factors-of-45-are-1-3-5-9-15-45-there-are-6-factors-the-probability-is-the-number-of-favorable-outcomes-divided-by-total-outcomes--37091182026-03-15T07:14:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-by-summing-the-numbers-and-dividing-by-the-count-20158992026-04-22T17:44:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-expressions-and-divide-by-3-81936542026-04-22T17:44:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-of-exactly-3-layers-allowing-flow-out-of-5-is-given-by-the-binomial-distribution-32192362026-04-22T17:52:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-of-at-least-2-reservoirs-being-below-average-is-1-minus-the-probabilities-of-0-or-1-reservoirs-being-below-average-85421522026-04-22T17:53:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-binomial-distribution-40387842026-04-22T17:53:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-number-of-integer-solutions--x-y--to-the-equation-65828402026-04-22T17:53:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-energy-produced-multiply-the-rate-of-energy-production-by-the-time-of-operation-34586022026-04-22T17:59:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-monthly-reduction-by-the-number-of-months-80231512026-04-22T17:59:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-find-the-first-and-last-4-digit-multiples-of-11-46398042026-04-22T17:59:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-5-out-of-12-plots-use-organic-farming-practices-and-the-scientist-selects-4-plots-at-random-we-want-the-probability-that-exactly-2-of-the-selected-plots-are-organic-29879912026-04-22T17:59:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-any-3-modules-from-8-30239152026-04-22T17:59:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-possible-4-character-passwords-with-no-restrictions-55556432026-04-22T17:59:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-count-the-number-of-valid-passwords-we-proceed-in-steps-42954282026-04-24T15:57:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-non-adjacent-selection-problem-we-model-the-8-layers-as-positions-1-to-8-we-want-to-choose-5-positions-such-that-no-two-are-consecutive-5127272026-04-24T15:57:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-6-questions-from-10-types-with-each-type-used-at-most-twice-and-total-count-exactly-6-we-count-the-number-of-integer-solutions-to-42184282026-04-24T15:57:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-three-digit-number-divisible-by-both-7-and-13-we-first-compute-their-least-common-multiple-since-7-and-13-are-both-prime-their-least-common-multiple-is-63382652026-04-22T18:02:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-radius--r--of-the-inscribed-circle-in-a-right-triangle-with-legs--a--and--b--and-hypotenuse--c--is-given-by-the-formula-52113072026-04-22T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-circle-is-inscribed-in-an-equilateral-triangle-and-has-diameter-10-cm-its-radius-is-25979322026-04-22T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-rectangle-have-width--2x--and-height--y--inscribed-in-a-semicircle-of-radius-5-with-its-base-on-the-diameter-the-top-corners-lie-on-the-semicircle-so-by-the-pythagorean-theorem-59176552026-04-22T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-sphere-of-radius--r--is-71793522026-04-22T18:02:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-probability-that-the-sum-of-four-6-sided-dice-equals-14-we-begin-by-determining-the-total-number-of-possible-outcomes-76524932026-04-22T18:03:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-hypergeometric-probability-problem-we-are-selecting-15-fish-from-a-population-of-200-of-which-10-are-tagged-we-want-the-probability-that-exactly-4-of-the-15-sampled-fish-are-tagged-96162452026-04-22T18:03:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-dividing-12-distinct-volunteers-into-3-unlabeled-teams-of-4-each-17015012026-04-22T18:03:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-three-numbers-is-even-if-70831472026-04-22T18:03:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-probability-that-at-least-2-of-the-4-selected-compounds-are-placebos-we-calculate-the-complement-of-the-probability-that-fewer-than-2-are-placebos-ie-0-or-1-placebo-81093962026-04-22T18:03:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-choosing-5-practices-from-9-total-with-4-coastal-and-5-non-coastal-19456692026-04-22T18:04:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-compute-the-total-profit-after-3-years-we-evaluate-both-profit-functions-at--x--3--and-sum-the-results-57319422026-04-24T15:58:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-slope--m--5308692026-04-24T15:58:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-how-many-drones-remain-undistributed-we-divide-120-by-8-and-find-the-remainder-36793772026-04-22T18:04:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-temperature-drop-we-add-the-two-drops-and-divide-by-2-95547482026-04-24T15:58:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-number-of-display-cases-we-determine-the-greatest-common-divisor-gcd-of-72-and-108-to-find-the-maximum-number-of-items-per-case-85761032026-04-24T15:58:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-number-of-swarm-units-we-compute-the-greatest-common-divisor-gcd-of-168-and-210-72159052026-04-24T15:58:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find--m--substitute--x--3--into-the-function-43077842026-04-22T18:04:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-48-and-64-factor-both-numbers-50952872026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term-a1--10-common-difference-d--5-and-last-term-an--50-63729272026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-factors-of-30-are-1-2-3-5-6-10-15-30-totaling-8-there-are-30-possible-integers-89401952026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-orthogonality-their-dot-product-must-be-zero-2x--3x---14--0-simplifying-5x---4--0-so-x--50300152026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-we-sum-the-lengths-and-divide-by-the-number-of-fish-66140392026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-of-pollinators-per-day-is-an-arithmetic-sequence-with-first-term--a--1--and-common-difference--d--3--the-formula-for-the--n--th-term-of-an-arithmetic-sequence-is-21079432026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-minimize-the-number-of-rectangles-we-should-maximize-the-area-of-each-rectangle-without-exceeding-15-square-inches-the-largest-area-per-rectangle-is-14-square-inches-a-possible-value-in-the-allowed-range-then-we-divide-the-total-area-by-this-value-30730332026-04-24T15:59:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-list-all-positive-divisors-of-60-the-prime-factorization-of-60-is-60927532026-03-15T07:48:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-prime-factor-of-143-we-test-divisibility-by-small-primes-27962132026-03-15T07:48:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-regular-cost-of-one-book-is-71907132026-03-15T07:48:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-red-to-yellow-marbles-is--53--this-means-for-every-5-red-marbles-there-are-3-yellow-marbles-25826882026-03-15T07:48:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-regular-cost-of-one-textbook-is-71945922026-03-15T07:48:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-square-has-sides-of-length-3-the-given-points-form-a-right-angle-at--1-4--with-the-fourth-vertex-completing-the-square-moving-from--1-1--to--1-4--up-3-units-then-right-3-units-to--4-4--the-missing-vertex-is-3-units-right-from--1-1--so--4-1--82531492026-04-24T16:00:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-triangle-have-sides--a--7---b--10---c--13--first-compute-the-area-using-herons-formula-the-semi-perimeter-is-5186952026-04-22T18:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-right-triangle-the-radius--r--of-the-circumscribed-circle-is-half-the-hypotenuse-91370712026-04-22T18:13:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-expression-for--r--is-undefined-when-the-denominator-is-zero-so-solve-42546852026-03-15T07:53:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--the-sum-of-the-first-three-terms-is-76157182026-04-22T18:13:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-a-of-a-regular-hexagon-with-side-length-s-is-given-by-83222842026-04-22T18:17:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-of-a-single-data-point-being-in-the-first-category-is--32892952026-04-22T18:24:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-multinomial-coefficient-the-number-of-ways-to-arrange-2-green-1-red-and-1-blue-49771692026-04-22T18:24:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-artifacts-in-the-five-exhibitions-be--a---2d-a---d-a-a--d-a--2d--forming-an-arithmetic-sequence-with-common-difference--d--the-total-number-of-artifacts-is-14917762026-04-22T18:24:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-visits-be--44209092026-04-22T18:24:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-cylindrical-part-is-given-by-the-formula-for-the-volume-of-a-cylinder-25835552026-04-22T18:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-solve-the-inequality-8140382026-04-22T18:28:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-find-the-greatest-distance-that-divides-18-24-and-30-evenly-this-is-the-greatest-common-divisor-gcd-of-the-three-numbers-76642522026-04-22T18:28:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-count-the-number-of-positive-divisors-of-360-93495192026-04-22T18:28:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-distinct-configurations-we-multiply-the-number-of-choices-for-grippers-by-the-number-of-choices-for-sensors-there-are-3-grippers-and-4-sensors-87276412026-04-24T16:05:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-6-steps-where-2-are-of-one-kind-catalysis-and-4-are-of-another-observation-with-indistinguishability-within-each-type-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-29034052026-04-24T16:04:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-order-of-the-excerpts-matters-we-compute-the-number-of-permutations-of-5-studies-chosen-from-8-this-is-given-by-97394492026-04-24T16:04:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-of-samples-that-contain-the-gene-multiply-the-total-number-of-samples-by-the-fraction-that-contains-the-gene-39774892026-04-22T18:34:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-number-of-initially-infected-people-by-the-average-number-of-new-infections-per-person-20974902026-04-22T18:34:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-daily-laying-rate-by-the-number-of-days-and-the-number-of-colonies-49326952026-04-22T18:33:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-total-number-of-positions-by-the-mutation-rate-51404672026-04-22T18:34:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-number-of-initially-infected-individuals-by--r0--to-find-the-number-of-second-generation-infections-66739142026-04-22T18:34:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-labor-rate-per-ant-by-the-number-of-ants-35870612026-04-22T18:34:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-total-number-of-sequences-by-the-fraction-that-contains-the-motif-35230642026-04-22T18:34:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-number-of-initially-infected-individuals-by-the-average-number-of-new-infections-per-person-46535892026-04-22T18:34:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-length-of-the-two-segments-we-add-the-two-lengths-together-and-then-divide-by-2-69728412026-04-22T18:34:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-found-by-summing-the-values-and-dividing-by-the-number-of-values-30130482026-04-22T18:34:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-be--r--and-the-height-be--h--39503862026-04-24T16:06:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-herons-formula-to-compute-the-area-4070912026-04-24T16:06:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-triangle-with-an-incircle-is-related-to-the-semiperimeter-and-inradius-by-48659942026-04-24T16:06:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--y--intercept-occurs-where--x--0--first-compute-the-slope--m--of-the-line-42574752026-04-24T16:06:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-system-of-equations-from-the-second-equation--x--y--2--substitute-into-the-first-8224432026-04-24T16:06:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--with--2x---1--in-the-original-function--fx--22402312026-04-22T18:37:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides-of-the-equation-24147472026-04-22T18:41:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-be--x--then-51011192026-04-22T18:41:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--4---y--21--into-the-equation-94422362026-04-22T18:41:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-4-samples-out-of-7-where-3-are-anomalous-call-them-type-a-and-4-are-normal-type-n-we-want-the-probability-that-at-least-2-of-the-4-selected-are-anomalous-85769532026-04-22T18:41:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-5-modules-from-8-46472732026-04-22T18:41:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-assign-6-distinct-drugs-to-6-patients-is-28569242026-04-22T18:41:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ordered-sequences-of-3-distinct-layers-from-5-is-90394402026-04-22T18:41:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-net-profit-per-delivery-is--12---150--1050--let--x--be-the-number-of-deliveries-96621492026-04-22T18:44:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-parameter-t-satisfy-82187272026-04-22T18:47:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-surface-in-spherical-coordinates-3457872026-04-24T16:09:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-sum-to-product-identity-96155332026-04-24T16:09:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-identity-for-tangent-addition-note-that-11471982026-04-24T16:09:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-distinct-sequences-is-the-multinomial-coefficient-accounting-for-indistinguishable-items-86848142026-03-15T08:32:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-load-we-sum-the-three-expressions-and-divide-by-3-6484912026-04-24T16:09:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-mean-sum-the-three-expressions-and-divide-by-3-26576042026-04-24T16:09:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term--a1--3--common-difference--d--5--and-last-term--an--48--4962852026-04-22T18:54:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-documents-is-4--3--2--9-since-the-documents-within-each-category-are-indistinguishable-the-number-of-distinct-orders-is-given-by-the-multinomial-coefficient-84976272026-04-22T18:54:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-samples-is-3--5--2--10-since-samples-within-each-category-are-indistinguishable-the-number-of-distinct-sequences-is-43466512026-04-22T18:54:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-individuals-is-4--5--3--12-since-individuals-within-each-status-category-are-indistinguishable-the-number-of-distinct-sequences-is-33104102026-04-22T18:54:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-euclidean-algorithm-33429302026-04-22T18:57:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--4--into-the-equation-5782472026-04-22T18:57:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--y--10--and--15x--20y--10x--25y--350--simplify-17162952026-04-22T18:57:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-half-life-we-add-the-two-half-lives-and-divide-by-2-83148792026-04-22T18:58:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-cost-of-one-headset-2261842026-04-22T18:58:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-and-simplifying-both-sides-63370782026-04-22T18:58:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-rate-per-minute-70722092026-04-22T18:58:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--r1x--r2x--96--at--x--5--first-add-the-two-functions-84461612026-04-22T18:58:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-a---b--1-express-b--a---1-substitute-into-aa--b--30-2014822026-04-22T19:01:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-y-intercept-set-x--0-78909992026-04-22T19:01:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-value-of--m--we-evaluate-both-functions-at--x--5--and-set-them-equal-34284342026-04-22T19:01:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-the-product--x--2x---5--76303242026-04-22T19:01:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--f2--g2--and-evaluate-both-sides-at--x--2--76195122026-04-22T19:01:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-consecutive-even-integers-be-2k-2k2-and-2k4-their-product-is-51184312026-04-22T19:02:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-integer-be-x-then-31783022026-04-22T19:02:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-and-width-of-the-rectangle-be--l--and--w--respectively-the-area-is--lw--100--and-the-perimeter-is--p--2l--w--to-minimize--p--the-rectangle-should-be-a-square-by-the-am-gm-inequality-or-calculus-so-set--l--w--70996512026-03-15T08:42:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-of-the-base-be--r--then-the-height--h--2r--the-volume--v--of-a-cone-is-85821802026-04-22T19:03:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-divisor-gcd-of-84-and-126-factorize-both-numbers-1686582026-04-22T19:07:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-number-of-mutated-microorganisms-among-the-20-new-samples-initially-45-of-100-microorganisms-show-mutations-27935062026-04-22T19:07:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initially-60-of-150-proposals-were-approved-55820392026-04-22T19:07:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initially-12-out-of-48-insects-are-resistant-56625172026-04-22T19:07:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-books-is-7--5--3--15-since-books-within-each-category-are-indistinct-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-95422882026-04-22T19:09:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-items-is-4--3--2--9-using-the-multinomial-coefficient-for-indistinct-items-32926512026-04-22T19:09:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-word-linguistics-has-11-letters-vowels-i-u-i-3-vowels-with-i-repeated-twice-consonants-l-n-g-s-t-c-s-7-consonants-with-s-repeated-twice-22384892026-04-22T19:09:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-partitioning-6-distinct-objects-into-4-non-empty-identical-boxes-which-uses-stirling-numbers-of-the-second-kind-the-value--s6-4--counts-the-ways-from-known-values--s6-4--65--10458122026-04-22T19:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-actual-length-of-the-trail-we-multiply-the-model-length-by-the-scale-factor-64849642026-04-22T19:16:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-we-use-the-formula-68670882026-04-22T19:16:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-forms-an-arithmetic-sequence-with-first-term--a--12--and-common-difference--d--5--the-general-term-is-72725552026-04-22T19:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-atomic-number-is-the-sum-of-protons-and-electrons-23271512026-04-22T19:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-subtract-the-fractions-we-find-a-common-denominator-48939202026-04-22T19:16:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rate-of-elevation-gain-is-31042702026-04-22T19:17:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-requires-finding-the-least-common-multiple-of-28-and-42-65207822026-04-22T19:16:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-the-map-length-by-the-scale-factor-98241622026-04-22T19:17:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--c--25--the-inradius--r--of-a-right-triangle-is-given-by-the-formula-60178532026-03-15T08:57:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-with-radius--r--is-88718502026-03-15T08:57:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-triangle-is-given-by-8553122026-03-15T08:57:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-positive-integers-less-than--50--that-are-multiples-of--7--are--7-14-21-28-35-42-49--there-are--7--such-numbers-the-sum-is-38630092026-04-24T16:14:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-sum-the-values-and-divide-by-the-number-of-months-59051832026-04-24T16:14:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-ensure-no-two-teachers-sit-next-to-each-other-we-first-fix-the-circular-arrangement-of-the-5-students-to-eliminate-rotational-symmetry-the-number-of-distinct-circular-arrangements-of-5-distinguishable-students-is-5-1--4--24-38714802026-03-15T08:57:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-terms-of-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--the-sum-of-the-first-and-fifth-terms-is-79782622026-04-22T19:19:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-speed-over--t--1--to--t--3--is-7305902026-04-24T16:15:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-30899342026-04-24T16:15:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-unrestricted-combinations-of-4-glaciers-from-9-99185832026-04-22T19:21:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio-of-red-to-yellow-marbles-is-75-this-means-for-every-7-red-marbles-there-are-5-yellow-marbles-59637372026-03-15T09:00:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-each-assignment-emilys-net-profit-is-75531762026-03-15T09:00:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate-the-numerical-values-of-the-bounds-77512262026-04-22T19:22:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-rectangular-urban-zone-is-inscribed-within-the-circular-satellite-coverage-the-diagonal-of-the-rectangle-is-equal-to-the-diameter-of-the-circle-the-rectangle-has-length-16-km-and-width-12-km-using-the-pythagorean-theorem-the-diagonal-d-is-89329592026-04-22T19:22:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-of-the-three-expressions--3v--2---5v---4--and--4v--7--we-add-them-together-and-divide-by-3-4223782026-04-22T19:22:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-starts-at-3-and-increases-by-4-each-time-85325562026-04-22T19:22:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-count-the-number-of-valid-sequences-of-length-5-using-3-distinct-symbols-where-no-two-adjacent-symbols-are-the-same-we-proceed-as-follows-64959692026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-binary-strings-of-length-6-each-bit-representing-a-city-being-infected-1-or-uninfected-0-such-that-no-two-adjacent-1s-appear-74380402026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-desired-probability-we-compute-the-number-of-favorable-outcomes-over-the-total-number-of-outcomes-28432322026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-ways-to-choose-3-dialects-from-6-51321862026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-combinatorics-problem-with-adjacency-constraints-each-region-can-be-in-one-of-3-states-c-o-or-r-however-adjacent-regions-cannot-have-the-same-state-67142872026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-integer-solutions-to-22998952026-04-22T19:24:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-inradius--r--of-a-right-triangle-can-be-expressed-using-the-formula-23024552026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-surface-area-of-both-the-sphere-and-cube-be--s--43576062026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diameter-of-the-inscribed-circle-is-equal-to-the-side-length--s--so-the-radius-is-91020142026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-consecutive-even-integers-be-x---2-x-and-x--2-their-sum-is-x---2--x--x--2--3x--96-solving-x--32-the-middle-stratum-has-x--32-which-is-the-greatest-of-the-three-since-x---2--30-x--32-x--2--34-27798222026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-is-the-sum-divided-by-5-76413572026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-the-average-of-the-three-expressions-is-computed-correctly-as-7594062026-04-22T19:31:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-92806612026-04-22T19:31:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-any-right-triangle-the-radius--r--of-the-inscribed-circle-can-be-expressed-using-the-formula-38170812026-04-22T19:32:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-we-first-compute-the-area-of-the-triangle-using-herons-formula-the-semiperimeter--s--is-85542762026-04-22T19:32:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-original-triangle-is-a-right-triangle-with-area-67373362026-04-22T19:32:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-recurrence-is-linear-and-homogeneous-80391162026-04-22T19:36:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-observe-that-the-right-hand-side-can-be-rewritten-as-77746792026-04-24T16:18:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-durations-and-divide-by-the-number-of-segments-3188432026-04-22T19:39:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-average-depth-sum-the-measurements-and-divide-by-the-number-of-measurements-63840162026-04-22T19:39:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-funding-amounts-and-divide-by-the-number-of-proposals-60080832026-04-22T19:39:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-possible-clip-length-determine-the-greatest-common-divisor-gcd-of-12-and-9-91502242026-04-22T19:39:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-average-temperature-sum-the-values-and-divide-by-the-number-of-measurements-46310542026-04-22T19:39:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-score-sum-the-scores-and-divide-by-the-number-of-proposals-7423952026-04-22T19:39:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-possible-clip-length-determine-the-greatest-common-divisor-gcd-of-144-and-108-convert-to-whole-numbers-by-multiplying-both-by-10-76779012026-04-22T19:39:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-average-salinity-sum-the-values-and-divide-by-the-number-of-samples-32629242026-04-22T19:39:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-path-is-the-difference-between-the-area-of-the-larger-circle-radius-12--2--14-meters-and-the-smaller-circle-radius-12-meters-24940812026-04-24T16:19:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-of-dosages-forms-an-arithmetic-progression-with-first-term--a--2--common-difference--d--3--and-last-term--l--200--using-the-formula-for-the--n--th-term-of-an-arithmetic-sequence-31283872026-04-22T19:44:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a--5--common-difference--d--6--and-last-term--l--299--using-the-formula-for-the--n--th-term-7981802026-04-22T19:44:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-red-pills-be--7x--and-yellow-pills-be--5x--given--7x--21--solve-for--x--48998172026-04-22T19:44:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--v--5k--where--k--is-a-positive-integer-then-9474962026-04-22T19:44:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-concentration-follows-an-arithmetic-sequence--a--3---d--4--we-want-the-smallest--n--such-that-15667072026-04-22T19:44:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-efficiency-follows-an-arithmetic-sequence--a--70---d--2--we-want-the-smallest--n--such-that-24633452026-04-22T19:44:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--a--4--and--b--3--into-the-expression--r--aa--b--50223832026-04-22T19:47:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-distributive-property-foil-method-to-expand-the-product-43678352026-04-22T19:47:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--a--5--into-the-equation-62006042026-04-22T19:47:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--1---y--4--and--a--3--into-the-equation--y--ax--b--81792302026-04-22T19:47:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-equation-in-spherical-coordinates-is-18589862026-04-22T19:49:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-observe-that-the-expression-is-a-perfect-square-trinomial-52783722026-04-22T20:03:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-two-complex-functions-6664192026-04-22T20:04:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-combinations-we-calculate-the-number-of-ways-to-choose-3-outcomes-from-8-and-2-groups-from-5-using-combinations-11200692026-04-22T20:05:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-choose-5-players-from-12-then-arrange-them-in-order-88435662026-04-22T20:05:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-total-combinations-44359892026-04-22T20:05:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-connected-loop-circular-chain-of--n--individuals-where-each-communicates-directly-with-two-neighbors-the-number-of-unique-communication-pairs-is-equal-to-the-number-of-edges-in-the-cycle-26267672026-04-24T16:25:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-at--t--3---p3--c3--first-compute--p3--64407842026-04-22T20:07:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-two-vectors-are-orthogonal-if-their-dot-product-is-zero-compute-the-dot-product-of-the-given-vectors-78972152026-04-22T20:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-eulers-formula-we-write-31363112026-04-22T20:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-triangle-is-given-by--t--87719452026-04-22T20:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-thread-forms-the-hypotenuse-of-a-right-triangle-where-one-leg-is-the-height--h--of-the-cylinder-and-the-other-leg-is-the-diameter--2r--of-the-circular-base-since-the-thread-spans-across-and-up-the-cylinder-using-the-pythagorean-theorem-the-length-of-the-thread-is-37728702026-04-22T20:07:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-calculate-the-number-of-valid-subsets-of-size-2-3-4-and-5-from-10-variables-where-no-two-are-adjacent-this-is-equivalent-to-counting-combinations-with-no-two-consecutive-elements-36909442026-04-22T20:08:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-coefficient-problem-the-total-number-of-distinct-sequences-is-the-number-of-permutations-of-9-items-where-4-are-identical-green-3-are-identical-blue-and-2-are-identical-red-86282422026-04-22T20:08:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-gcd-takes-the-minimum-exponent-for-each-common-prime-factor-80843642026-04-22T20:08:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-triangle-with-base--b--4--and-height--h--is--53447382026-04-22T20:08:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-each-of-the-5-distinct-water-samples-is-analyzed-on-a-different-day-and-the-order-matters-this-is-a-permutation-problem-the-number-of-distinct-sequences-in-which-5-unique-items-can-be-ordered-is-5-77121332026-04-22T20:11:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-determine-the-number-of-distinct-sequences-of-length-7-where-4-are-crystalline-c-and-3-are-gelatinous-g-this-is-a-combinatorics-problem-involving-permutations-of-multiset-59062782026-04-22T20:11:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-a-sequence-of-8-terrain-classifications-with-5-indistinguishable-mountainous-m-and-3-indistinguishable-plateau-p-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-82785422026-04-22T20:11:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-given-by-48697052026-04-22T20:15:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-radius--r--of-the-inscribed-circle-in-an-equilateral-triangle-with-side-length--a--is-given-by--r--79917882026-04-22T20:20:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-perimeter-of-both-shapes-be--p--for-the-square-side-length-is--69359552026-04-22T20:20:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-vectors-to-be-orthogonal-their-dot-product-must-equal-zero-compute-the-dot-product-29247732026-04-22T20:20:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-of-the-rectangle-be--l--and-the-width-be--w--the-perimeter-is-given-by-59215812026-04-22T20:22:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-rationalize-the-denominator-multiply-the-numerator-and-denominator-by-the-conjugate-of-the-53349172026-04-22T20:24:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-factors-of-60-the-prime-factorization-of-60-is-19526462026-04-24T16:29:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-for--x--begin-with-the-equation-99474322026-04-24T16:29:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-an-arithmetic-progression-the-terms-are-12693792026-04-24T16:29:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-divisor-gcd-of-144-and-216-we-factorize-both-numbers-44219232026-04-24T16:29:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-10-odd-numbers-are-1-3-5-7-9-11-13-15-17-19-their-sum-is-44139642026-04-22T20:24:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-on-nous-demande-de-calculer-73151032026-04-22T20:30:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-polynomial-equation-of-the-form-59531082026-04-22T20:31:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-simplify--fx--for--x-82871272026-04-22T20:31:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-sides-of-the-rectangle-be--x--and--y--we-are-given-37750952026-04-22T20:31:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-3-engineers-as-a-single-block-total-blocks-to-arrange-5666482026-04-22T20:35:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-distinct-arrangements-is-calculated-using-the-multinomial-coefficient-there-are-a-total-of-4--3--2--9-symbols-with-repetitions-of-each-type-the-formula-is-50562392026-03-15T10:15:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-permutation-of-multiset-objects-the-total-number-of-rows-is-5--4--3--12-the-number-of-distinct-sequences-is-given-by-60639462026-03-15T10:15:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-nanobots-is-6--2--1--9-the-identical-items-repair-and-communication-reduce-the-count-via-the-multinomial-coefficient-78417312026-03-15T10:15:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-identify-the-prime-factors-of-the-expression-the-given-expression-is-already-factored-into-primes-47991522026-04-24T16:32:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-daily-generation-for-all-three-panels-35243242026-04-24T16:32:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-sequences-in-which-5-identical-fern-spores-4-identical-moss-spores-and-3-identical-lichen-fragments-can-be-arranged-over-12-days-one-per-day-since-items-within-each-category-are-indistinguishable-this-is-a-multinomial-coefficient-problem-the-total-number-of-distinct-arrangements-is-given-by-12351162026-04-22T20:37:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-permutations-of-9-characters-where-there-are-3-identical-xs-2-identical-ys-and-4-identical-zs-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-63276212026-04-22T20:37:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-apply-the-cauchy-schwarz-inequality-in-the-engel-form-titus-lemma-34790232026-04-22T20:41:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-largest-angle-we-apply-the-law-of-cosines-the-longest-side-is-13-cm-so-we-let--a--7---b--10--and--c--13--and-compute-angle--c--opposite-side--c--22664042026-04-24T16:34:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-spherical-to-cartesian-conversion-formulas-8814962026-04-24T16:34:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-base-triangle-lies-in-the--xy--plane-and-is-equilateral-verified-by-equal-side-lengths-of-4-the-fourth-vertex-lies-directly-above-the-centroid-of-the-base-triangle-the-centroid--g--is-1519182026-04-22T20:45:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-lies-on-the-line-and-on-the-perpendicular-from--34--to-the-line-the-slope-of-the-line-is-2-so-perpendicular-slope-is----35966482026-04-24T16:34:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-partition-1000-distinguishable-instances-into-groups-of-sizes-700-training-200-validation-and-100-test-we-use-multinomial-coefficients-the-number-of-ways-to-assign-the-data-is-65780022026-04-24T16:34:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-geometric-progression-where-each-generation-doubles-the-number-of-new-infections-the-initial-case-generation-0-infects-2-people-in-generation-1-each-of-whom-infects-2-more-and-so-on-for-5-generations-the-total-number-of-infected-people-is-the-sum-of-a-geometric-series-30471492026-04-22T20:46:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x-be-the-amount-of-minerals-using-the-ratio-25--10x-solve-for-x-44552302026-04-22T20:46:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-4-from-both-sides-95225612026-04-22T20:46:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-parts-are-7--3--10-the-time-for-coding-is-18167912026-04-22T20:46:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term-a--5-common-difference-d--4-and-last-term-l--41-76211452026-04-22T20:46:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-shelves-are-identical-we-are-counting-the-number-of-ways-to-partition-a-set-of-6-distinguishable-elements-into-at-most-4-non-empty-unlabeled-subsets-this-is-equivalent-to-summing-the-stirling-numbers-of-the-second-kind--sn-k--for--k--1--to--4--where--n--6--1443392026-04-22T20:49:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-species-from-5-is-3893032026-04-22T20:49:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-2-flavors-from-8-is-1494422026-04-22T20:49:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-math-books-as-one-block-and-science-books-as-another-there-are--2--ways-to-arrange-the-blocks-math-first-or-science-first-within-each-block-the-books-can-be-permuted-26070872026-04-22T20:49:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-first-determine-the-total-number-of-ways-to-choose-3-layers-from-the-5-which-is-given-by-the-combination-51045462026-04-22T20:50:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-choose-4-out-of-the-6-cultural-practices-which-can-be-done-in-11854262026-04-22T20:50:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-5-samples-from-12-3-per-period-is-67920922026-04-22T20:50:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-3-vials-from-5-is-21097812026-04-22T20:50:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-4-tribes-from-7-is-16162962026-04-22T20:50:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-for--x--from-the-given-equation-36573352026-04-22T20:50:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-tangent-addition-formula-24046572026-04-22T20:50:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-lets-simplify-the-expression-98267252026-04-22T20:50:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-recurrence-problem-let-an-be-the-number-of-valid-sequences-of-length-n-with-no-two-consecutive-es-using-letters-e-and-s-15478592026-04-24T16:35:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d-a--3d--where--a--is-the-fourth-term-and--d--is-the-common-difference-the-sum-of-the-first-three-terms-is-16365962026-04-22T20:57:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-first-term-be--a--and-the-common-ratio-be--r--then-12625482026-04-22T20:57:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-sum-of-the-first-four-terms-substitute--n--1-2-3-4--into-the-formula-24768992026-04-22T20:57:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-sequence-be--a---2d-a---d-a-a--d-a--2d--87083232026-04-22T20:57:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-two-digit-number-such-that-51446002026-04-22T21:03:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-rate-of-change-of-the-volume-with-respect-to--x--we-compute-the-derivative-of--v--with-respect-to--x--58271322026-04-24T16:39:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-rate-of-absorption-compute--at--12673602026-04-24T16:39:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with-the-expression-for--dn1---dn--27801442026-04-24T16:39:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-numbers-be--a1-a2-a3-a4-a5--from-the-given-we-have-63232442026-04-24T16:40:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-number-of-successful-new-investments-originally-she-funded-20-companies-with-a-65-success-rate-84182172026-04-22T21:10:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-geometric-series-with-first-term--a--50--common-ratio--r--12--and--n--5--terms-the-sum-is-42534922026-04-22T21:10:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-isolating-one-radical-97131542026-04-22T21:10:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-area-of-a-triangle-given-coordinates-of-its-vertices-70269572026-04-22T21:15:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-be--r--a-perpendicular-from-the-center-to-a-chord-bisects-the-chord-so-half-the-chord-is--5--cm-and-the-perpendicular-distance-from-the-center-is--6--cm-using-the-pythagorean-theorem-in-the-right-triangle-formed-by-the-radius-half-the-chord-and-the-perpendicular-64516382026-04-22T21:15:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-cube-be--s--the-space-diagonal--d--of-a-cube-satisfies-45925182026-04-22T21:15:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-weekly-growth-values-be--a1-a2-a3-a4-a5--where-each--38875132026-04-22T21:15:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-given-values--c0--200---c--50---t--1000--66227762026-04-22T21:29:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-be-x-then-we-are-given-36435182026-04-22T21:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-recursive-sequence-44892242026-04-22T21:44:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-three-expressions-is-their-sum-divided-by-3-so-54785352026-04-22T21:47:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-population-doubles-every-3-hours-we-use-exponential-growth-18821962026-04-22T21:47:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-over-140-days-the-number-of-sheds-is-85787282026-04-22T21:47:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-lengths-and-divide-by-the-number-of-snakes-31296692026-04-22T21:47:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-convert-6-hours-to-seconds-45610102026-04-22T21:47:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-express--1--i--and--1---i--in-polar-form-94692722026-04-24T16:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-vertex-form-of-a-quadratic-equation-is-42751822026-04-22T22:02:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-total-retreat-in-30-years-first-calculate-the-annual-retreat-rate-95688452026-03-15T11:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-rate-of-travel-per-minute-97852152026-03-15T11:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-a-proportion-based-on-the-given-ratio-63087712026-03-15T11:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-rate-per-hour-30011092026-03-15T11:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-how-many-3-hour-intervals-are-in-24-hours-54181432026-03-15T11:51:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-the-three-flow-rates-is-given-by-68844642026-04-22T22:13:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-values-39717572026-04-22T22:13:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-two-columns-are-identical-and-each-has-height--h--and-radius--r--the-volume-of-one-cone-is-given-by-the-formula--v--70496712026-04-24T16:56:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-original-side-length-of-an-equilateral-triangle-be--s--the-area--a--of-an-equilateral-triangle-is-given-by-98362026-04-24T16:56:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-least-common-multiple-lcm-of--48--and--72--first-factor-both-numbers-58430112026-04-24T16:56:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-tasked-with-partitioning-7-distinguishable-soil-samples-into-3-non-empty-indistinguishable-groups-this-is-equivalent-to-computing-the-stirling-number-of-the-second-kind--s7-3--which-counts-the-number-of-ways-to-partition-7-labeled-objects-into-3-non-empty-unlabeled-subsets-90567552026-04-22T22:26:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-8-distinguishable-grants-to-4-distinguishable-laboratories-with-each-lab-receiving-at-least-one-grant-this-is-a-problem-of-counting-surjective-onto-functions-from-a-set-of-8-elements-to-a-set-of-4-elements-87373082026-04-22T22:26:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-how-much-memory-is-freed-33841042026-04-22T22:28:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-values-and-divide-by-the-number-of-data-points-3590112026-04-22T22:30:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-arctic-foxes-be-x-the-ratio-47-implies-3249712026-04-22T22:30:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-particle-comes-to-rest-when-its-velocity-is-zero-ie-when--ht--0--71943982026-04-22T22:30:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-sequence-be--a---2d-a---d-a-a--d-a--2d--where--a--is-the-third-term-and--d--is-the-common-difference-69651312026-04-22T22:30:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-slant-height--l--of-a-right-circular-cone-can-be-found-using-the-pythagorean-theorem-since-the-height-radius-and-slant-height-form-a-right-triangle-10748632026-04-22T22:30:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-triangle-given-two-sides--a--and--b--and-the-included-angle--heta--is-5639762026-04-22T22:30:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--fx--y--fxfy--for-all-real--x-y--and--f1--2--this-is-the-multiplicative-functional-equation-over-the-reals-first-lets-compute--f2--22367222026-03-15T12:10:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-three-points-on-the-quadratic-function-99982832026-03-15T12:10:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-line-be-parameterized-as-4458182026-03-15T12:10:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-velocity-vector-32397602026-03-15T12:10:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-vertex-rewrite-the-equation-in-vertex-form-by-completing-the-square-32786652026-03-15T12:10:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--p--be-the-cost-of-a-plastic-sheet-and--b--the-cost-of-a-bamboo-panel-the-system-of-equations-is-67152042026-03-15T12:11:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--be-the-time-in-hours-after-species-y-departs-species-x-has-traveled--t---15--hours-the-distance-difference-is-54107952026-03-15T12:11:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-triangle-be--3k--and--4k--so-the-hypotenuse-is-98085732026-04-24T16:59:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-an-equilateral-triangle-with-side-length--s-13481872026-03-15T12:23:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-equation-in-spherical-coordinates-13194242026-04-24T17:00:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-sequence-be-37872472026-04-24T17:00:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-time-per-instrument-is--x--x--10--2x--10--for-120-instruments-the-total-time-is--1202x--10--3300--expanding-66003602026-04-24T17:00:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-sensor-readings-in-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d--given-that-the-sum-of-the-first-and-fifth-readings-is-40-40220492026-04-22T22:50:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--e--aa--b--ba--b--factor-out--a--b--92685902026-04-22T22:50:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert-the-base-eight-number--3218--to-base-ten-we-use-the-place-value-expansion-74246792026-04-24T17:00:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-formula-for-the-sum-of-the-first--n--squares-is-63745002026-04-24T17:00:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-lets-define--an--as-the-number-of-valid-forecasts-for--n--days-where-no-two-consecutive-days-are-rainy-3496742026-04-22T22:55:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-animals--4--3--7--all-are-distinguishable-95222592026-04-24T17:02:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-regular-tetrahedron-with-edge-length--a--is-given-by-85353562026-03-15T12:34:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-cm-first-compute-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter--s--is-96962362026-03-15T12:36:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-9-patent-applications-4-from-company-a-indistinguishable-among-themselves-3-from-company-b-and-2-from-company-c-we-are-to-count-the-number-of-distinct-sequences-in-which-these-can-be-reviewed-where-only-the-company-of-origin-matters-not-individual-applications-and-indistinguishability-within-companies-59563352026-03-15T12:36:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-sequences-of-length-6-where-there-are-2-low-2-medium-and-2-high-transmission-days-since-the-days-are-ordered-over-time-and-the-counts-per-category-are-fixed-this-is-a-permutation-of-a-multiset-83791862026-03-15T12:36:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-remainder-theorem-again-we-evaluate--f-1--since-the-divisor-is--u--1--u----1--47996522026-03-15T12:40:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that--f3--g3--first-compute--f3--345462026-04-24T17:05:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-first-term-be--a--and-the-common-difference-be--d--54204492026-03-15T12:56:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-value-of--m--such-that--fx--and--gx--intersect-at-exactly-one-point-set--fx--gx--99870452026-04-24T17:07:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find--k--such-that--px--qx--has-exactly-one-solution-set-22778902026-04-24T17:08:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-functional-equation-is-43254092026-04-22T23:23:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-permutations-of-6-chimpanzees-is-6--720-to-count-the-number-of-arrangements-where-kali-and-momo-are-adjacent-treat-them-as-a-single-entity-this-gives-5-arrangements-and-kali-and-momo-can-be-ordered-in-2-ways-so-2-imes-5--240-subtracting-from-the-total-720---240--480-21596032026-04-22T23:25:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-permutations-of-5-monkeys-is-5--120-in-half-of-these-aki-precedes-bongo-in-the-other-half-bongo-precedes-aki-so-the-number-of-sequences-where-aki-does-not-precede-bongo-is--39166892026-04-22T23:25:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-herons-formula-to-find-the-area-of-the-triangle-first-compute-the-semi-perimeter-51651642026-04-22T23:27:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-area-of-a-triangle-in-terms-of-its-inradius--r--and-semi-perimeter--s--14652332026-04-22T23:27:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-area-formula-for-an-equilateral-triangle-67682242026-04-24T17:10:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-both-numbers-into-primes-78235932026-04-22T23:32:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-geometric-series-with-first-term--a--200--common-ratio--r--085--and--n--4--terms-28861932026-04-24T17:11:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-net-profit-per-delivery--18---2--16--dollars-71335782026-04-24T17:11:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-with-the-initial-amount-of-disinfectant-which-is-3-liters-subtract-the-amount-used-75804852026-04-24T17:11:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-smallest-4-digit-number-is-1000-and-the-largest-is-9999-we-need-to-find-how-many-numbers-between-1000-and-9999-inclusive-are-divisible-by-11-37010452026-04-24T17:11:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-10-positive-even-integers-are-2-4-6-8-10-12-14-16-18-20-59420192026-04-22T23:32:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-cost-of-one-kit-be-k-76133372026-04-22T23:39:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term-a1--3-common-difference-d--4-and-last-term-an--31-80624292026-04-22T23:39:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-solving-the-compound-inequality-75159782026-04-22T23:41:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-the-number-of-distinct-sequences-in-which-the-chemist-can-use-5-catalysts---a-b-c-d-e---with-each-used-exactly-3-times-over-15-days-where-the-order-of-reactions-matters-but-reactions-with-the-same-catalyst-are-indistinguishable-33710942026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-7-items-where-87362992026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-12-items-with-repeated-types-4236822026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-note-that-the-sum-of-three-numbers-is-even-if-and-only-if-82828962026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-items-4-red-r-3-green-g-2-blue-b-with-indistinguishability-within-colors-66238632026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-10-items-with-57364462026-04-22T23:46:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-expression-1608472026-04-22T23:49:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from-a---b--2-express-a--b--2-84493342026-04-22T23:56:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-initial-height-of-the-balloon-be--h--and-the-horizontal-distance-traveled-be--x--from-the-station-the-initial-tangent-of-the-angle-of-elevation-gives-50416212026-04-23T00:00:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-of-squares-formula-is--51549022026-04-24T17:18:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-number-of-ways-to-choose-3-different-flavors-from-8-this-is-a-combination-68456942026-04-24T17:18:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-tetrahedron-all-edges-are-equal-compute--ab---ac---bc--81076522026-04-23T00:04:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-there-is-no-restriction-on-overlap-as-species-and-zones-are-distinct-categories-we-calculate-the-combinations-separately-and-multiply-72759302026-04-23T00:08:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-involves-arranging-6-identical-dna-samples-and-4-identical-rna-samples-the-total-number-of-items-is-6--4--10-and-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-6827792026-04-23T00:10:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-partition-n-distinguishable-objects-into-k-indistinguishable-boxes-is-given-by-the-sum-of-stirling-numbers-of-the-second-kind-sn1--sn2-for-n7-and-k2-34562952026-04-23T00:10:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-circular-arrangements-with-alternating-roles-fix-one-scientists-position-to-account-for-rotational-symmetry-the-remaining-4-scientists-can-be-arranged-in-4-ways-and-the-3-engineers-can-be-arranged-in-3-ways-thus-23595762026-04-23T00:10:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-items-is-4--3--2--9-with-repetitions-the-number-of-distinct-arrangements-is-given-by-46427262026-04-23T00:10:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-prime-factorization-of-180-29737722026-03-15T13:48:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-ice-thickness-measurements-be-represented-by--7x--and--4x--where--7x--28--solving-for--x--we-divide-both-sides-by-7-34010362026-03-15T13:48:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-pottery-shards-be--5x--and-the-number-of-stone-tools-be--3x--the-total-number-of-artifacts-is-given-as-65538942026-04-24T17:20:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-by-summing-the-values-and-dividing-by-the-number-of-values-59496392026-04-24T17:20:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-use-the-euclidean-algorithm-start-by-dividing-the-larger-number-by-the-smaller-number-and-take-the-remainder-30478542026-04-24T17:20:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-6-distinguishable-compost-mixtures-into-4-indistinguishable-plots-with-no-plot-left-empty-this-is-equivalent-to-finding-the-number-of-ways-to-partition-a-set-of-6-labeled-elements-into-exactly-4-non-empty-unlabeled-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted--sn-k--we-seek--s6-4--94472422026-04-23T00:14:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-partitioning-7-distinguishable-birds-into-exactly-3-non-empty-indistinguishable-subsets-migration-routes-treated-as-unlabeled-this-is-again-a-stirling-number-of-the-second-kind--s7-3--37353002026-04-23T00:14:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-distance-we-add-the-two-distances-and-divide-by-2-29060052026-04-23T00:14:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-load-time-we-add-the-two-times-and-divide-by-2-57753622026-04-23T00:14:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-we-add-the-two-sizes-and-divide-by-2-99255652026-04-23T00:14:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-we-add-the-two-distances-and-divide-by-2-4618562026-04-23T00:14:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-as-63061632026-04-23T00:14:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-determine-the-domain-67968782026-03-15T13:59:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-semi-perimeter--s--66120482026-03-15T13:59:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-verify-the-triangle--9--10--17--compute-semi-perimeter--s--52344492026-03-15T13:59:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-involves-arranging-7-sensors-where-3-are-identical-temperature-sensors-2-are-identical-motion-sensors-and-2-are-identical-cameras-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-16439012026-04-24T17:25:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-polymer-uses-7-components-3-identical-microcapsules-3-identical-vascular-channels-and-1-unique-healing-agent-the-number-of-distinct-sequences-is-calculated-by-74836382026-04-24T17:25:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-student-reads-7-items-4-identical-math-books-2-identical-lab-notebooks-and-1-unique-journal-the-number-of-distinct-sequences-is-70597592026-04-24T17:25:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-a-bee-visits-5-flowers-in-random-order-and-we-are-to-compute-the-probability-that-flowers-a-b-and-c-are-visited-in-increasing-order-ie-a-before-b-b-before-c-not-necessarily-consecutively-the-bee-chooses-its-next-flower-uniformly-at-random-from-unvisited-ones-since-all-permutations-of-the-5-flowers-are-equally-likely-due-to-the-uniform-random-selection-assuming-no-memory-bias-we-consider-all-5-permutations-of-the-flowers-20199462026-04-24T17:27:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-where-the-first-term--a--3--the-common-difference--d--4--and-the-last-term--l--99--the-formula-for-the--n--th-term-of-an-arithmetic-sequence-is-10338822026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-month-of-the--n--th-grant-is-given-by-19396782026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term--a--2--common-difference--d--4--the-sum-of-the-first--n--terms-is-given-by-20264402026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with--a--5---d--8--and--x--100--the--n--th-term-is-19330172026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-geometric-sequence-with--a--3---r--3--the-sum-of-the-first--n--terms-is-37874222026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-circumference-of-the-circle-in-which-the-isosceles-triangle-is-inscribed-we-first-determine-the-triangles-circumradius-the-triangle-has-sides-a--5-b--5-and-c--6-using-the-formula-for-the-circumradius-r-of-a-triangle-78050632026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-triangle-have-legs-a-b-and-hypotenuse-z-the-inradius-c-of-a-right-triangle-is-86063352026-04-23T00:35:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-and-width-of-the-rectangle-be--x--and--y--respectively-the-perimeter-is-given-by-81009412026-04-23T00:35:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-radius-be--r--and-the-height-be--h--the-total-surface-area-of-a-cylinder-is-22013052026-04-23T00:35:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-primero-identifica-los-primos-menores-que-60-hay-17-primos-91181812026-04-23T00:40:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-people--6--4--10-for-circular-arrangements-of--n--distinguishable-people-the-number-of-distinct-seating-arrangements-is--n-1--this-accounts-for-rotational-symmetry-78111652026-04-23T00:40:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-applications-is-65122012026-04-24T17:30:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-unknown-leg-be--x--using-the-pythagorean-theorem-17266212026-04-23T00:55:44+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-check-if-the-triangle-is-right-verify-if-the-pythagorean-theorem-holds-assume--25--is-the-hypotenuse-then-18175172026-04-23T00:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor--2025--we-find-50952532026-04-23T01:02:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-odd-integers-be--n-n2-n4-n6--where--n--is-odd-since-they-are-consecutive-odd-numbers-they-are-all-relatively-prime-in-pairs-we-analyze-the-divisibility-properties-11459442026-04-23T01:02:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-be--x--we-are-given-32378662026-04-23T01:02:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-five-composite-numbers-are-4-6-8-9-10-we-compute-their-prime-factorizations-40012672026-04-23T01:04:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-a--2025-and-b--2020-we-use-the-identity-79158522026-04-24T17:36:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-values-and-divide-by-3-80825782026-04-23T01:17:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-equation-for-the-mean-52882052026-03-15T14:54:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-investments-and-divide-by-3-19587502026-03-15T14:54:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-volumes-and-divide-by-3-91804392026-03-15T14:54:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--3456--to-base-ten-we-expand-it-as-8345552026-03-15T14:54:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-the-equation--5z---7--3z--4--expand-the-right-hand-side--5z---7--3z--12--subtract--3z--from-both-sides--2z---7--12--add-7-to-both-sides--2z--19--divide-by-2--z--53507242026-04-23T01:18:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-samples-is-given-by-dividing-total-time-by-sampling-interval--69831982026-04-23T01:18:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-ratio--75--means-for-every-7-red-neurons-there-are-5-blue-neurons-let--x--be-the-number-of-blue-neurons-then--76137062026-04-23T01:18:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-rotations-required-for-alignment-is-determined-by-the-least-common-multiple-lcm-of-the-number-of-teeth-divided-by-the-teeth-count-of-the-gear-the-lcm-of-15-and-20-is-60-for-the-smaller-gear-15-teeth-the-number-of-rotations-is--59716142026-04-23T01:18:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-alice-and-bob-each-choose-a-uniform-random-time-between-100-pm-and-200-pm-to-send-a-message-and-we-are-given-the-condition-that-alice-sends-after-bob-we-are-to-compute-the-conditional-probability-that-bob-sent-his-message-before-130-pm-given-this-information-80533372026-04-24T17:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-compute-the-probability-that-exactly-two-out-of-four-fair-6-sided-dice-show-a-number-greater-than-4-91147862026-04-24T17:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-compute-the-number-of-distinct-permutations-of-the-letters-in-the-word-mathematics-85064232026-04-24T17:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-begin-by-identifying-the-total-number-of-ways-to-choose-4-stations-from-the-10-available-which-is-given-by-the-combination-87281492026-04-23T01:25:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-ways-to-choose-4-districts-from-12-18041512026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-problem-in-combinatorics-involving-derangements-we-are-to-find-the-probability-that-exactly-2-of-6-distinct-elements-are-fixed-in-their-original-positions-65898262026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a--3--common-difference--d--4--and-last-term--l--99--the-formula-for-the--n--th-term-of-an-arithmetic-sequence-is-34969632026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-of-three-expressions-is-the-sum-divided-by-3-854732026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-bees-be--7x--and-the-number-of-butterflies-be--5x--given--7x--35--49899052026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a--2--common-difference--d--4--and-last-term--l--198--98610552026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with--a--5---d--7--and--l--96--97423272026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-beetles-be--8x--and-the-number-of-ladybugs-be--3x--given--8x--24--61355582026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-years-be--x--and--y--with--x--y--we-have-the-system-14502012026-04-24T17:42:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-clay-percentage-be--c--then-the-sand-percentage-is--c--15--79381722026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-weekly-increase-be--d--the-first-week-2000-views-54340782026-04-24T17:42:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-triangle-with-sides--a--13---b--14--and--c--15--the-longest-side-is--c--15--so-we-are-to-find-the-shortest-altitude-to-this-sidethis-corresponds-to-the-altitude-from-the-vertex-opposite-the-15-km-side-61014502026-04-24T17:42:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-right-triangle-have-legs--a---b--and-hypotenuse--z--the-area--a--is-68844142026-04-24T17:42:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-arrange-6-distinct-items-a1-a2-stone-axes-b1-b2-ceramic-bowls-c1-c2-gold-pectorals-with-the-restriction-that-no-two-items-of-the-same-type-are-adjacent-although-the-tools-are-distinct-they-are-grouped-by-type-with-two-identical-looking-items-per-type-but-physically-distinct-due-to-being-unique-artifacts-91711682026-04-23T01:28:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--mu--u---36125532026-04-23T01:28:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-is-given-by--54468612026-04-24T17:43:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-divisor-is-quadratic-the-remainder-will-be-of-the-form--ax--b--we-write-12864092026-04-24T17:43:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--p1--p2--p3--0--the-polynomial-has-roots-at--x--1-2-3--therefore-38651432026-04-24T17:43:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-a-line-to-a-given-point-is-found-by-projecting-the-point-perpendicularly-onto-the-line-18765252026-04-24T17:44:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-apply-the-method-of-lagrange-multipliers-and-inequalities-to-minimize--fx-y-z--59408102026-04-23T01:32:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-slope-m-48840952026-04-23T01:40:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-rectangle-is-inscribed-in-the-circle-its-diagonal-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-the-length-of-the-diagonal-d-is-15417242026-04-23T01:42:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-calculate-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-s-is-86885292026-04-23T01:42:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-isosceles-triangle-with-equal-sides-of-10-cm-and-base-12-cm-the-altitude-to-the-base-divides-it-into-two-equal-segments-of-6-cm-each-using-the-pythagorean-theorem-we-find-the-height-h-72250972026-04-23T01:42:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-angular-velocity-we-use-the-formula-27665972026-04-23T01:42:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-number-of-identical-square-tiles-that-can-exactly-cover-a-12-by-16-rectangular-region-we-need-to-determine-the-largest-square-tile-size-that-can-evenly-divide-both-dimensions-this-is-the-greatest-common-divisor-gcd-of-12-and-16-72433032026-04-23T01:43:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-stars-in-the-second-region-be--x--the-ratio-of-stars-is--75--and-the-first-region-has-35-stars-we-set-up-a-proportion-63119272026-04-23T01:43:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-participants-from-the-first-neighborhood-be--4x--and-from-the-second-be--9x--the-total-number-of-participants-is-38184012026-04-23T01:43:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-of-radius--r--is-30059352026-04-23T01:43:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-an-equilateral-triangle-the-circumradius--r--is-related-to-the-side-length--s--by-the-formula-11666722026-04-23T01:43:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-total-of-7-distinct-ice-fractures-and-the-glaciologist-selects-4-of-them-at-random-we-are-to-compute-the-probability-that-a-specific-critical-fracture-say-fracture-a-is-among-the-4-selected-33527052026-04-23T01:43:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-there-are-8-distinct-fish-species-and-3-are-chosen-at-random-we-want-the-probability-that-both-the-blue-damselfish-species-b-and-the-spurgery-species-s-are-among-the-3-selected-47774852026-04-23T01:43:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--f--be-the-cost-per-cup-of-flour-and--s--the-cost-per-cup-of-sugar-the-system-of-equations-is-44657362026-04-24T17:47:55+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-initial-preferential-responses--060-imes-200--120--669532026-04-24T17:47:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from--a---b--1--express--a--b--1--substitute-into--2a--3b--10--37442502026-04-24T17:47:57+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-we-calculate-the-total-number-of-4-digit-ids-and-the-number-of-favorable-outcomes-where-exactly-two-distinct-digits-appear-22961822026-04-23T01:48:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-compute-the-desired-probability-we-begin-by-determining-the-total-number-of-ways-to-choose-4-combinations-from-12-27272312026-04-23T01:52:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-mixtures-from-9-is-52406732026-04-23T01:52:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-trials-from-7-is-89853992026-04-23T01:52:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-5-settings-from-10-is-78562402026-04-23T01:52:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-4-materials-from-15-is-23470702026-04-23T01:52:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-3-studies-from-8-is-77197652026-04-23T01:52:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-list-the-factors-of-100-1-2-4-5-10-20-25-50-there-are-8-factors-55379822026-04-24T17:50:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-expressions-5w--2--3w---1--2w--4--10w--5-17963692026-04-24T17:50:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-lines-intersect-at--x---2--substitute--x---2--into-both-equations-and-equate-the--y--values-10051302026-04-23T02:05:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--y--100---x--into--ux-y--59121662026-04-23T02:09:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-dot-product-must-be-zero-5207042026-04-23T02:09:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sum-telescopes-as-follows-10231972026-04-23T02:19:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-time-is-calculated-by-dividing-distance-by-speed-15249342026-04-23T02:19:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-path-consists-of-two-segments-a-o-c-and-c-o-b-the-total-distance-is-15327182026-04-23T02:19:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-number-of-combinations-we-use-combinations-for-the-flavors-and-simple-multiplication-for-selecting-one-syrup-80391662026-04-23T02:20:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--fx--has-degree-less-than-2-we-can-write-it-as--fx--ax--b--using-the-given-values-25267132026-04-23T02:23:06+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-instantaneous-rate-of-change-of--pt--is-given-by-its-derivative-10414232026-03-15T15:59:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-subtract-2-from-both-sides-to-bring-all-terms-to-one-side-17157982026-03-15T15:59:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-18-and-24-factorize-both-numbers-14967472026-04-23T02:24:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-sensors-are-indistinct-in-physical-position-the-robots-configuration-depends-only-on-how-many-sensors-are-in-each-mode-not-their-spatial-arrangement-this-is-a-problem-of-counting-the-number-of-non-negative-integer-solutions-to-the-equation-29799942026-04-23T02:24:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-partitioning-7-distinguishable-widgets-into-4-non-empty-unlabeled-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-s7-4-and-since-the-zones-are-indistinguishable-we-do-not-multiply-by-any-permutation-factor-from-known-values-90193162026-04-23T02:24:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-ways-to-choose-5-bees-from-12-is-84336742026-04-23T02:24:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-coefficient-problem-we-are-arranging-8-actions-where-3-are-of-one-kind-3-of-another-and-2-of-a-third-all-indistinct-within-types-the-number-of-distinct-sequences-is-41101992026-04-23T02:24:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-number-of-ways-to-choose-4-insects-from-10-4736062026-04-23T02:24:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-fossils-is-5--4--3--12-and-we-are-arranging-a-sequence-of-12-items-where-5-are-of-one-type-dinosaur-4-of-a-second-type-reptile-and-3-of-a-third-type-insect-with-indistinguishability-within-each-type-the-number-of-distinct-arrangements-is-given-by-the-multinomial-coefficient-92019142026-04-23T02:24:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-area-enclosed-by-the-curve-54071602026-03-15T16:07:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-arithmetic-mean-we-sum-the-integers-and-divide-by-the-number-of-integers-87331252026-03-15T16:08:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-circular-arrangement-fix-one-6988772026-04-23T02:39:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-all-three-servers-synchronize-compute-the-least-common-multiple-lcm-of-18-24-and-30-99782752026-04-23T02:51:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-out-the-greatest-common-divisor-gcd-of-the-coefficients-which-is-3-72225172026-04-24T18:11:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-equations-equal-to-solve-for-x-55676602026-04-24T18:11:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-y-intercept-occurs-when-x--0-substituting-49663382026-04-24T18:11:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-follows-a-binomial-distribution-with-parameters--n--6--and--p--4410092026-04-23T03:04:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-binomial-probability-formula-with--n--7---p--29065342026-04-23T03:04:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--x--is-a-positive-multiple-of-6-we-can-write--x--6k--where--k--is-a-positive-71862972026-04-23T03:04:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-budgets-and-divide-by-the-number-of-grants-45119042026-04-24T18:13:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-depths-and-divide-by-3-39681782026-04-23T03:10:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-up-the-equation-for-the-average-53070192026-04-23T03:10:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-seed-cones-be-x-the-ratio-53-implies--59600512026-04-23T03:15:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum--14--18--22--16--20--90-mean---26536382026-04-23T03:15:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-sum-the-expressions-4u--1--2u--5--6u---3--12u--3-divide-by-3--82519322026-04-23T03:15:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-funding-amounts-and-divide-by-the-number-of-projects-35565072026-04-24T18:19:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-total-trees-28--36--42--54--160-the-smallest-prime-factor-of-160-is-2-as-160-is-even-15314312026-04-24T18:19:22+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--v--is-a-multiple-of-6-lets-try-successive-multiples-of-6-and-cube-them-50518332026-04-24T18:19:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-centroid-g-of-triangle-abc-2836142026-04-24T18:20:43+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-distinct-soundscapes-we-calculate-the-number-of-ways-to-choose-3-ambient-layers-from-8-and-2-effects-from-5-since-the-order-does-not-matter-we-use-combinations-31084572026-04-23T03:50:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-4-proteins-from-12-is-given-by-84734972026-04-23T03:50:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-2-interpretations-from-7-is-93798952026-04-23T03:50:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-order-of-the-effects-matters-and-no-repetition-is-allowed-we-are-looking-for-the-number-of-permutations-of-10-effects-taken-4-at-a-time-35871772026-04-23T03:50:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-ways-to-choose-3-drugs-from-9-is-27740242026-04-23T03:50:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-choose-5-participants-from-10-85459692026-04-23T03:50:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-distance-from-a-point--x0-y0--to-a-line--ax--by--c--0--is-given-by-the-formula-70981862026-04-23T03:52:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-recall-that-in-spherical-coordinates-10552042026-04-23T03:53:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-shortest-altitude-we-first-note-that-the-shortest-altitude-corresponds-to-the-longest-side-which-is-15-cm-we-begin-by-computing-the-area-of-the-triangle-using-herons-formula-the-semi-perimeter-is-8872672026-04-23T03:52:58+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-original-equilateral-triangle-be--s--the-area-of-an-equilateral-triangle-is-9614062026-04-23T03:53:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-triangle-have-legs--a---b--and-hypotenuse--c--25--the-inradius--r--5--in-a-right-triangle-the-inradius-is-given-by-92404392026-04-23T03:53:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-triangle-be-isosceles-with-sides--a--10---b--10---c--12--the-semi-perimeter-is-64103822026-04-23T03:53:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-overall-average-word-length-we-compute-the-weighted-average-based-on-the-number-of-words-in-each-manuscript-55412742026-04-23T03:53:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-square-region-is-4367232026-04-23T03:53:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-list-all-prime-numbers-less-than-20-86186192026-04-23T03:53:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-complex-number-is-in-polar-form-25187512026-04-23T03:53:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-diameter-add-the-two-measurements-and-divide-by-2-78464292026-04-24T18:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-cost-of-one-airplane-34167332026-04-24T18:26:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-3-gemstones-as-a-single-super-object-this-gives-6--4--1--11-items-to-arrange-with-the-6-pottery-shards-and-4-statues-being-distinct-the-number-of-arrangements-is--89246492026-04-23T03:56:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-fraction-calibrated-is--34878212026-03-15T17:32:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-rational-functions-the-horizontal-asymptote-depends-on-degrees-of-numerator-and-denominator-here-both-are-degree-1-so-the-asymptote-is-y--67128132026-03-15T17:32:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-minimum-distance-from-the-origin-to-the-surface-occurs-when--24294332026-04-23T03:56:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-choosing-3-viral-strains-from-7-with-4-enveloped-e-and-3-non-enveloped-n-we-want-the-probability-that-exactly-2-are-enveloped-88616742026-03-15T17:34:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-5-animals-where-there-are-2-identical-males-m-2-identical-females-f-and-1-juvenile-j-the-number-of-distinct-permutations-of-a-multiset-is-given-by-66783042026-03-15T17:34:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-lets-define--an--as-the-number-of-valid-sequences-of-length--n--using-symbols-l-m-h-with-no-two-consecutive-hs-82937512026-03-15T17:34:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-rna-segments-where-4-are-identical-of-type-a-3-of-type-b-and-2-of-type-c-the-number-of-distinct-permutations-is-97548502026-03-15T17:34:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-6-events-3-identical-grooming-g-2-identical-play-p-and-1-aggression-a-the-number-of-distinct-permutations-is-33235682026-03-15T17:34:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--an--be-the-number-of-valid-sequences-of-length--n--with-no-two-consecutive-high-h-readings-81573102026-03-15T17:34:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-ensure-diversification-the-vectors-must-be-orthogonal-so-their-dot-product-is-zero-95961532026-04-24T18:27:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-both-terms-27817022026-04-24T18:27:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-sum-of-the-numbers-2394622026-04-23T04:08:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-subgroup-sizes-be--2n1-2n3-2n5-2n7--these-are-four-consecutive-odd-numbers-68486132026-04-23T04:08:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-inscribed-circle-is-86872362026-04-23T04:08:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-note-that--1---x--y--z---1---y--x--z---1---z--x--y--so-the-expression-becomes-31974602026-04-23T04:08:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set--x--0--in--2x--3y--6--92261882026-04-23T04:17:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-72-and-168-we-first-perform-prime-factorization-3531272026-04-23T04:19:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-count-the-total-number-of-positive-integers-less-than-or-equal-to-60-which-is-60-64469362026-04-23T04:19:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-finding-the-greatest-common-factor-of-24-and-36-97634822026-03-15T17:56:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-requires-the-least-common-multiple-of-15-and-21-3681142026-03-15T17:56:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-the-least-common-multiple-of-3-and-5-43379802026-03-15T17:56:15+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-product-26836432026-04-24T18:35:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-lengths-and-divide-by-2-83878492026-03-15T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-mean-sum-the-expressions-and-divide-by-3-86070112026-03-15T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-each-number-78752026-03-15T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-add-the-lengths-and-divide-by-2-31198832026-03-15T18:02:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-reflection-of-a-point-across-the-plane--z--0--simply-negates-the--z--coordinate-since-the-bird-passes-through--2-2-4--its-reflection-lies-at--2-2--4--the-parabolas-axis-being-along-the--z--axis-confirms-the-symmetry-and-the-reflection-is-consistent-with-geometric-inversion-through-the-plane-38154052026-03-15T18:02:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-velocity-vector-is-the-derivative-of-the-position-vector-83446732026-03-15T18:02:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-radius--r--of-the-circle-can-be-found-using-the-formula-for-the-distance-from-the-center-to-a-chord-69407072026-04-23T04:29:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-a-regular-hexagon-inscribed-in-a-circle-the-side-length-equals-the-radius--r--thus--r--3--cm-the-circumference--c--is-58351082026-04-23T04:29:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the--y--intercept-set--x--0--in-the-equation-98599092026-04-23T04:29:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-3-related-theories-as-a-single-block-this-reduces-the-problem-to-arranging-4-entities-the-block--3-other-theories-around-a-circular-table-which-has-4-1--6-arrangements-the-3-related-theories-within-the-block-can-be-permuted-in-3--6-ways-total-arrangements-11430362026-04-23T04:29:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-stars-and-bars-problem-with-the-constraint-that-each-project-gets-at-least-one-grant-first-allocate-1-grant-to-each-project-leaving-7---5--2-grants-to-distribute-freely-the-number-of-non-negative-integer-solutions-to-x1--x2--x3--x4--x5--2-is-42756952026-04-23T04:29:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-number-of-successful-trials-in-the-additional-10-attempts-the-total-successful-trials-are--065-imes-40--x--26--x--the-overall-success-rate-is--96316142026-04-23T04:29:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--p--represent-the-number-of-user-profiles-processed-in-one-test-run-since-each-test-run-uses-128-mb-and-processes-16-profiles-the-memory-per-profile-is--16136392026-04-23T04:29:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--s--represent-the-number-of-simulations-that-can-run-simultaneously-since-each-simulation-uses-32-gb-of-ram-the-total-ram-required-is--32s--the-total-available-ram-is-512-gb-so-we-solve-36074352026-04-23T04:29:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--n--represent-the-number-of-data-entries-since-each-entry-contributes-005-seconds-the-total-time-is--005n--the-constraint-is-97036342026-04-23T04:29:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--s--represent-the-number-of-simulations-each-simulation-requires-2-cpus-and-8-gb-ram-so-the-constraints-are-70798732026-04-23T04:29:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-8-and-14-we-first-find-the-prime-factorization-of-each-number-97739132026-04-23T04:34:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-when-both-processes-align-we-compute-the-least-common-multiple-lcm-of-12-and-18-82390512026-04-23T04:34:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-the-asteroids-align-we-compute-the-least-common-multiple-lcm-of-15-and-25-25635272026-04-23T04:34:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-reactions-align-we-compute-the-least-common-multiple-lcm-of-10-and-14-27465692026-04-23T04:34:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-when-both-notifications-align-we-compute-the-least-common-multiple-lcm-of-9-and-15-31120392026-04-23T04:34:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-crater-patterns-align-we-compute-the-least-common-multiple-lcm-of-6-and-9-69463582026-04-23T04:34:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-functions-execute-together-we-compute-the-least-common-multiple-lcm-of-6-and-10-33719982026-04-23T04:34:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-rationalize-the-denominator-multiply-numerator-and-denominator-by-the-conjugate-of-the-denominator-50324262026-04-23T04:34:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-value-of--x--that-makes-the-two-vectors-orthogonal-we-compute-their-dot-product-and-set-it-equal-to-zero-65648342026-04-24T18:37:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-define--fu--based-on-the-given-functional-equation-given-47578722026-04-24T18:37:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--pp--q--45--and--p--q--9--65050952026-04-23T04:38:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-number-of-successful-trials-in-the-additional-10-the-initial-success-rate-is--065-imes-40--26--successful-attempts-after-10-more-trials-total-attempts-are--50--and-total-successes-are--26--x--the-new-success-rate-is--64862262026-04-23T04:38:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-project-bs-allocation-then-project-a-receives--x--40000--the-equation-is--x--x--40000--240000--simplify--2x--200000--so--x--100000--59432632026-04-23T04:38:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-base-nine-number--2439--means-19439922026-04-23T04:38:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-calculate-the-total-number-of-ways-to-select-and-arrange-5-manuscripts-from-10-ensuring-that-at-least-2-are-astronomy-related-we-consider-three-valid-cases-exactly-2-astronomy-manuscripts-exactly-3-and-exactly-4-53237622026-04-23T04:38:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--o--be-the-cost-of-an-oxygen-tank-and--w--the-cost-of-a-water-purifier-the-system-of-equations-is-46608002026-03-15T18:14:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--be-the-legs-per-antelope-and--z--the-legs-per-zebra-given--a--4--and--z----the-total-legs-are-53451442026-03-15T18:14:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-from--p--q--7--express--q--7---p--substitute-into-the-first-equation-81964922026-03-15T18:14:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-herpetologist-is-observing-9-distinct-species-5-frogs--4-snakes-and-each-species-is-observed-exactly-once-over-9-days-since-all-species-are-distinct-and-order-matters-the-number-of-sequences-is-simply-the-number-of-permutations-of-9-distinct-items-61671902026-03-15T18:14:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-8-signals-where-there-are-3-ss-3-ls-and-2-ps-since-the-total-number-of-signals-is-8-and-some-are-repeated-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-53659672026-03-15T18:14:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-7-episodes-where-3-are-on-physicists-p-2-on-chemists-c-and-2-on-biologists-b-since-videos-within-each-field-are-indistinguishable-the-number-of-distinct-arrangements-is-68354552026-03-15T18:14:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-7-sensors-3-ts-2-ss-and-2-ds-since-sensors-of-the-same-type-are-indistinguishable-the-number-of-distinct-sequences-is-51442202026-03-15T18:14:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-6-videos-with-2-from-each-of-3-topics-calculus-c-geometry-g-number-theory-n-the-number-of-distinct-sequences-is-15374832026-03-15T18:14:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-area-using-herons-formula-38904522026-04-23T04:39:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-verify-that-the-triangle-is-valid-using-the-triangle-inequality-14324872026-04-24T18:39:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-simplify-the-left-hand-side-77369722026-04-23T05:04:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-expand-both-sides-of-the-equation-2978192026-04-23T05:04:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--10--into-the-formula-41810382026-04-23T05:05:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-velocity-is-the-derivative-of-position-89546362026-04-23T05:05:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the--t--intercept-occurs-when--d--0--38502642026-04-23T05:05:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--m--be-the-cost-of-a-lab-manual-and--g--be-the-cost-of-a-safety-goggles-the-system-of-equations-is-65722132026-04-24T18:46:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--be-the-cost-per-cubic-meter-of-a-and--b--for-b-the-system-is-28306282026-04-24T18:46:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--p--be-the-cost-of-a-plastic-bottle-and--s--for-a-soda-case-the-system-is-38906922026-04-24T18:46:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-groups-are-indistinguishable-and-each-must-receive-at-least-one-project-the-problem-reduces-to-partitioning-6-distinct-projects-into-4-non-empty-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind--s6-4--the-formula-for--sn-k--is-24058642026-04-23T05:10:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-3-modules-where-7-are-distinguishable-prototypes-and-5-are-identical-backup-units-since-the-prototype-units-are-unique-and-the-backup-units-are-indistinguishable-we-consider-combinations-based-on-how-many-prototypes-are-selected-from-0-to-3-and-how-many-backup-units-are-used-up-to-3-but-limited-by-availability-and-total-draw-90032942026-04-23T05:14:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-selecting-one-indistinguishable-stem-cell-line-from-4-identical-types-so-only-one-choice-for-the-stem-cell-category-and-three-distinct-neural-templates-from-6-distinguishable-ones-since-the-stem-cell-line-is-identical-across-all-selecting-any-one-gives-one-unique-type-72382262026-04-23T05:14:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--fx--is-a-cubic-polynomial-its-third-differences-are-constant-we-compute-the-values-3360192026-04-23T05:17:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-binomial-probability-formula-with--n--6---p--03--and--k--2--6584772026-04-23T05:20:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-12-15-and-20-we-begin-by-factoring-each-number-into-its-prime-components-60812422026-04-23T05:20:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-system-of-congruences-41821442026-04-23T05:20:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-smallest-integer-greater-than-100-that-is-a-common-multiple-of-18-and-24-but-not-a-multiple-of-30-first-compute-the-least-common-multiple-lcm-of-18-and-24-13510612026-04-23T05:20:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-when-both-cycles-align-we-compute-the-least-common-multiple-lcm-of-36-and-60-27275442026-04-23T05:20:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-compute-the-probability-that-exactly-2-out-of-5-students-receive-their-originally-assigned-experiment-a-fixed-set-of-permutations-with-the-remaining-3-elements-forming-a-derangement-with-no-fixed-points-among-the-other-students-4491922026-04-23T05:20:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-maximum-population-we-first-compute-the-derivative-of--pt--42465002026-04-23T05:24:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-progression-rate-is--ln--4116092026-04-23T05:24:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--h--3--into-the-function-46622272026-04-23T05:24:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--5--into-the-function-55583242026-04-23T05:24:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-recognize-this-as-a-perfect-square-trinomial-50556652026-04-23T05:24:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-a-circle-is-proportional-to-the-square-of-its-radius-84078092026-04-23T05:24:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-right-triangle-can-be-written-as-11940072026-03-15T18:59:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-combinatorics-problem-with-restrictions-based-on-categories-the-investor-must-choose-3-startups-each-from-a-different-research-area-the-available-areas-are-84485742026-03-15T19:04:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-permutations-of-5-projects-where-52907952026-03-15T19:04:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-6-initiatives-with-indistinguishability-within-categories-26072152026-03-15T19:04:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-sum-the-values-and-divide-by-the-number-of-phases-72422402026-04-23T05:31:23+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-12-distinct-zones-to-12-stages-4-of-one-type-high-pressure-3-of-another-injectant-and-5-of-a-third-monitoring-since-the-zones-are-distinguishable-but-stages-of-the-same-type-are-indistinct-in-labeling-the-number-of-distinct-arrangements-is-the-multinomial-coefficient-10605572026-04-24T18:51:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-treat-the-2-specific-concepts-that-must-be-adjacent-as-a-single-block-this-reduces-the-problem-to-arranging-6-items-the-5-other-concepts-and-the-1-block-these-6-items-can-be-arranged-in-6-ways-within-the-block-the-2-concepts-can-be-arranged-in-2-ways-therefore-the-total-number-of-arrangements-is-17182192026-04-24T18:52:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-of-radius-r-is-80767192026-04-23T05:33:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--px--is-a-cubic-polynomial-its-third-differences-are-constant-we-compute-the-values-and-successive-differences-30499012026-04-23T05:36:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-19-and-23-are-both-prime-numbers-their-least-common-multiple-lcm-is-their-product-97273122026-04-23T05:36:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-when-all-three-cycles-align-we-compute-the-least-common-multiple-lcm-of-18-24-and-30-43679042026-04-23T05:41:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-13-and-17-are-both-prime-and-distinct-their-least-common-multiple-is-simply-their-product-91541472026-04-23T05:41:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-1-from-both-sides--46561932026-04-23T05:43:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-temperature-we-add-the-two-temperature-readings-and-divide-by-2-65862092026-04-24T18:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-is-the-number-of-favorable-outcomes-years-divisible-by-5-divided-by-the-total-number-of-outcomes-total-years-80548652026-04-24T18:55:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-find-the-prime-factorization-of-84-76851222026-04-24T18:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-add-the-two-fractions-we-find-a-common-denominator-69314002026-04-24T18:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-is-the-number-of-favorable-outcomes-months-divisible-by-3-divided-by-the-total-number-of-outcomes-total-months-42501832026-04-24T18:55:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-probability-is-the-number-of-favorable-outcomes-volcanoes-on-plate-boundary-divided-by-the-total-number-of-outcomes-total-volcanoes-34680522026-04-24T18:55:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-calculate-the-desired-probability-we-consider-the-favorable-arrangements-where-all-3-17th-century-discoveries-are-adjacent-and-divide-by-the-total-number-of-possible-arrangements-of-the-10-discoveries-30158462026-04-23T05:47:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-sphere-with-radius-x-is-56472252026-04-23T05:48:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-identity-ana--b--95164352026-04-23T05:48:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given--lu--u---22776062026-04-23T05:48:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--pn--represent-the-frog-population-after--n--months-the-population-grows-by-8-monthly-so-each-month-it-is-multiplied-by-108-and-then-150-frogs-are-lost-starting-with--p0--2000--the-recurrence-relation-is-79553982026-04-23T05:54:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-original-side-length-is--s--new-side-length-is--s---3--the-area-of-a-regular-hexagon-is--60049332026-04-23T05:56:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-largest-perfect-square-divisor-of-2025-18438772026-04-23T06:03:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-find-the-largest-perfect-square-that-divides-1024-38436062026-04-23T06:03:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-radius--r--of-the-inscribed-circle-in-any-triangle-is-given-by-87676442026-04-23T06:03:46+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-estimate-the-limit-we-find-the-fixed-point-of--fx--x---49709522026-04-23T06:08:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-gears-is-6--5--4--15-we-want-the-probability-that-exactly-2-of-the-3-selected-gears-are-blue-84186592026-04-23T06:08:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-rolling-an-8-sided-die-four-times-we-want-exactly-30670532026-04-23T06:08:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-total-number-of-distinct-permutations-of-a-multiset-is-given-by-dividing-the-factorial-of-the-total-number-of-letters-by-the-product-of-the-factorials-of-the-counts-of-each-repeated-letter-here-there-are-7-letters-in-total-18924132026-04-23T06:16:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-code-fixes-4-digits-d-e-f-and-permutes-only-a-b-c-in-the-remaining-3-positions-the-total-number-of-permutations-of-a-b-c-is-3--6-among-these-the-number-of-permutations-where-a-comes-before-b-is-exactly-half-since-in-half-the-arrangements-a-precedes-b-and-in-half-b-precedes-a-the-cases-are-symmetric-61530032026-04-23T06:16:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-altitude-corresponds-to-the-longest-side-which-is--15--m-first-compute-the-area-using-herons-formula-the-semiperimeter--s--is-32417822026-04-23T06:16:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-the-full-sphere-of-radius--y--is-68245212026-04-23T06:16:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-permutation-of-a-multiset-9-total-items-with-repetitions-the-number-of-distinct-sequences-is-4372712026-04-23T06:17:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-distances-in-arithmetic-sequence-be--a---2d-a---d-a-a--d-3767632026-04-23T06:17:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-equation-in-cylindrical-coordinates-26775082026-04-23T06:20:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-sum-of-the-four-numbers-82357662026-04-23T06:20:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-four-consecutive-odd-integers-are--11071-11073-11075-11077--their-sum-is-49163022026-04-23T06:20:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-four-consecutive-odd-integers-be--n-n2-n4-n6--where--n--is-odd-the-product-is-34857312026-04-23T06:21:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert--27538--to-base-ten-expand-using-powers-of-8-90863692026-04-23T06:21:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-intersection-set-the-two-equations-equal-9475772026-04-23T06:30:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-evaluate--f3--and--g3--15215702026-04-23T06:30:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-solve-for--x--in-the-equation--2x--3--11--41558242026-04-23T06:30:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-y-intercept-occurs-when--x--0--substitute--x--0--into-the-equation-83042692026-04-23T06:30:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-two-equations-equal-14638872026-04-23T06:30:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-6-distinguishable-fish-species-into-4-indistinguishable-zones-with-each-zone-receiving-at-least-one-species-this-is-equivalent-to-finding-the-number-of-ways-to-partition-a-set-of-6-labeled-elements-into-exactly-4-non-empty-unlabeled-subsets-this-is-given-by-the-stirling-numbers-of-the-second-kind-denoted--sn-k--here--n--6---k--4--1136712026-04-23T06:34:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-sequences-of-length-7-using-symbols-w-m-c-such-that-3081502026-04-23T06:34:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-hypergeometric-probability-problem-we-have-8516572026-04-23T06:34:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-ways-to-distribute-8-distinguishable-fish-into-3-distinguishable-zones-a-b-c-with-22884972026-04-23T06:34:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-5-distinct-transitions-occurring-at-unique-seconds-in-a-3600-second-interval-with-all-orderings-of-transition-times-uniformly-random-we-are-to-compute-the-probability-that-two-specific-transitions-say-t1-and-t2-occur-at-consecutive-seconds-in-the-sequence--that-is-one-immediately-follows-the-other-either-t1-then-t2-or-t2-then-t1-88290032026-04-23T06:34:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-distinct-circular-arrangements-of-12-pollen-grains--5-oak-o-4-pine-p-and-3-birch-b--such-that-no-two-birch-grains-are-adjacent-all-grains-of-the-same-type-are-indistinguishable-arrangements-are-considered-up-to-rotation-circular-symmetry-37648172026-04-23T06:34:30+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-compute-the-probability-that-in-a-random-assignment-of-difficulty-levels-e-m-h-to-7-tasks-exactly-2-are-hard-and-no-two-hard-tasks-are-adjacent-62613992026-04-24T19:06:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-following-system-of-congruences-38908542026-04-23T06:36:25+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-compute-the-difference-of-squares-94771912026-04-23T06:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-square-root-of-a-square-gives-the-absolute-value-81506302026-04-23T06:54:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-cost-per-unit-is-the-total-cost-divided-by-the-number-of-units-29332092026-04-23T06:54:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-mutation-appears-every-15th-strand-so-it-occurs-at-multiples-of-15-75974152026-04-23T06:54:04+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-fourth-vertex-be--d--x-y-z--and-denote--a--000---b--220---c--202--in-a-regular-tetrahedron-all-edges-have-equal-length-14015112026-04-23T06:55:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-sequence-be--a---3d-a---d-a--d-a--3d--9705312026-04-24T19:12:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-three-given-vertices-be-a--100-b--010-c--001-and-let-d--xyz-be-the-fourth-vertex-of-a-regular-tetrahedron-so-that-all-edges-have-equal-length-26547232026-03-15T20:44:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-square-both-sides-to-eliminate-the-square-root-22925832026-04-23T07:14:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-horizontal-asymptote-compare-the-degrees-of-the-numerator-and-denominator-both-are-degree-1-so-the-horizontal-asymptote-is-the-ratio-of-the-leading-coefficients-17407342026-04-23T07:14:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--2--into-both-functions-57677252026-04-23T07:31:36+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--x--5--into--fx--53191602026-04-23T07:31:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-speed-is-the-magnitude-of-the-velocity-vector-calculated-as-2888472026-04-23T07:31:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-hypotenuse--z--for-any-right-triangle-the-inradius--c--is-given-by-56898292026-04-23T07:31:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-rectangle-is-inscribed-in-the-circle-so-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-using-the-pythagorean-theorem-69303932026-04-23T07:31:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-asked-to-count-the-number-of-distinct-ways-to-assign-responses-to-100-participants-such-that-40-say-yes-35-say-no-and-25-say-unsure-since-the-participants-are-distinguishable-but-we-are-only-counting-the-number-of-distinct-groupings-ie-how-many-ways-to-divide-the-100-people-into-three-labeled-groups-of-fixed-sizes-this-is-a-multinomial-coefficient-problem-the-number-of-such-response-patterns-is-given-by-96723942026-04-23T07:32:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-assigning-one-of-3-priority-levels-say-a-b-c-to-each-of-7-distinct-signals-in-sequence-with-the-restriction-that-no-two-adjacent-signals-have-the-same-priority-the-first-signal-has-3-choices-for-each-subsequent-signal-since-it-cannot-have-the-same-priority-as-the-previous-one-there-are-2-choices-therefore-the-total-number-of-valid-assignments-is-93106152026-04-23T07:32:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-a-sequence-of-8-video-segments-with-fixed-counts-3-animations-a-2-simulations-s-2-data-overlays-d-and-1-live-action-l-since-the-segments-of-the-same-type-are-indistinct-in-effect-though-filmed-separately-we-are-counting-the-number-of-distinct-permutations-of-a-multiset-the-number-of-distinct-editing-orders-is-70417772026-04-23T07:32:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-regular-hexagon-inscribed-in-a-circle-the-side-length-equals-the-radius-of-the-circle-31493262026-04-23T07:32:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-18-24-and-36-we-first-perform-prime-factorization-87300252026-04-23T07:34:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-probability-we-first-count-the-positive-divisors-of-120-that-are-less-than-or-equal-to-60-the-prime-factorization-of-120-is-24218372026-04-23T07:34:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-factor-numerator-and-denominator-13862122026-03-15T21:15:39+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-local-minima-first-compute-the-derivative-52932002026-04-23T07:36:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-polynomial-58865342026-04-23T07:36:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-each-term-in-the-first-polynomial-by-each-term-in-the-second-60526402026-04-23T07:38:27+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-mass-of-one-astrolabe-be--m--kilograms-since-there-are-three-identical-astrolabes-their-total-mass-is--3m--this-is-given-as-27-kilograms-65137972026-04-23T07:41:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-sequence-be--a-a--d-a--2d-a--3d-a--4d--5200352026-04-23T07:43:59+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-tout-dabord-convertissons-3-livres-en-kilogrammes--42608712026-04-23T07:56:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-volume-formula--v--59809972026-04-23T07:56:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-five-weights-be--551092026-04-23T07:56:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-largest-square-plot-size-we-compute-the-greatest-common-divisor-gcd-of-72-and-48-17269122026-04-24T19:27:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term--a--3--common-difference--d--5--and-last-term--l--198--36016032026-04-23T07:59:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by--x---1x--1--to-eliminate-denominators-99485622026-04-23T08:00:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-composition--fgx--is-defined-when--gx--is-defined-and-when--gx--lies-in-the-domain-of--f--since--fx--is-a-polynomial-its-defined-for-all-real-numbers-so-we-only-need--gx--to-be-defined-71429632026-04-23T08:00:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-the-expression-under-the-square-root-must-be-non-negative-80712292026-04-23T08:00:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-given--x--y--z--1--note-that--y--z--1---x---z--x--1---y---x--y--1---z--so-the-expression-becomes-7975642026-04-23T08:00:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-solve-this-first-compute-each-square-and-sum-them-79255662026-04-23T08:37:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-least-common-multiple-of-12-and-18-76611902026-04-23T08:37:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-treat-each-species-as-a-single-block-there-are-3-ways-to-arrange-the-3-blocks-chimpanzees-gorillas-orangutans-within-each-block-the-animals-can-be-permuted-among-themselves-5-for-chimpanzees-3-for-gorillas-and-4-for-orangutans-the-total-number-of-arrangements-is-86983182026-04-23T08:37:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-multinomial-problem-where-we-count-the-number-of-permutations-of-a-multiset-the-total-number-of-sequences-is-78051932026-04-23T08:37:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-two-digit-number-be--n--we-are-given-84022602026-04-23T08:40:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--n--be-the-two-digit-number-such-that-20396562026-04-23T08:40:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-no-two-adjacent-blocks-can-be-the-same-color-and-we-have-equal-numbers-of-red-and-blue-blocks-the-only-possible-alternating-patterns-are-rbrbrb-and-brbrbr-these-are-the-only-two-sequences-that-satisfy-the-condition-and-use-3-red-and-3-blue-blocks-no-other-arrangement-avoids-adjacent-duplicates-for-example-starting-with-b-leads-to-brbrbr-and-starting-with-r-leads-to-rbrbrb-no-other-starting-point-avoids-conflict-thus-there-are-exactly-2-valid-arrangements-86241532026-04-23T08:42:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area--a--of-a-right-triangle-is-given-by--a--7397642026-04-23T08:42:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume-of-a-hemisphere-of-radius--r--is-33441022026-04-23T08:42:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-semiperimeter--s--of-the-triangle-is-62349282026-04-23T08:42:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-volume--v--of-a-cone-is--37662832026-04-23T08:42:53+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-length-and-width-of-the-rectangle-be--l--and--w--the-perimeter-is-35814672026-04-23T08:42:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-smallest-positive-integer--n--such-that-69011712026-04-23T08:44:37+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-system-of-three-dot-product-equations-78907952026-04-24T19:42:07+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-side-length-of-the-original-equilateral-triangle-be--s--the-area--a--of-an-equilateral-triangle-is-given-by-9274992026-04-23T09:02:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-right-triangle-have-legs--a---b--and-hypotenuse--z--the-area--a--of-the-triangle-is-61187352026-04-23T09:02:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-average-ph-we-sum-the-three-expressions-and-divide-by-3-80057902026-03-15T22:45:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-is-the-sum-divided-by-3-66963452026-03-15T22:45:31+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-average-is-the-sum-divided-by-2-28919142026-03-15T22:45:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-us-define-a-substitution-to-simplify-the-expression-set-50452832026-04-23T09:08:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-an-equilateral-triangle-the-altitude-from-one-vertex-to-the-opposite-side-splits-the-triangle-into-two-congruent-right-triangles-using-the-pythagorean-theorem-the-altitude--h--satisfies-99930902026-04-23T09:08:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-right-triangle-the-radius--r--of-the-circumscribed-circle-is-half-the-length-of-the-hypotenuse-first-compute-the-hypotenuse--c--68517332026-04-23T09:08:12+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-closest-point-on-the-line--y--2x--1--to--3--2--is-found-by-minimizing-the-distance-squared-38366932026-04-23T09:10:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-12-simulations-where-each-of-the-4-scenarios-appears-exactly-3-times-this-is-a-multinomial-coefficient-problem-the-number-of-distinct-sequences-is-the-multinomial-coefficient-87000522026-04-23T09:11:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-fx-is-a-cubic-polynomial-its-third-differences-are-constant-compute-the-values-23671422026-04-23T09:11:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-v3-integrate-vt-89848742026-04-23T09:11:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-particle-is-at-rest-when-vt--st--0-40300122026-04-24T19:46:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-cauchy-schwarz-inequality-in-the-engel-form-titus-lemma-30936172026-04-24T19:46:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-number-of-distinct-arrangements-is-calculated-by-dividing-the-factorial-of-the-total-number-of-items-by-the-factorial-of-the-number-of-identical-items-here-there-are-5-total-permutations-but-since-2-charts-are-identical-we-divide-by-2-thus-the-number-of-distinct-arrangements-is--57594782026-04-24T19:48:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-convert-the-base-5-number-32145-to-base-10-expand-it-using-powers-of-5-73998852026-04-24T19:48:08+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-time-both-signals-align-is-the-least-common-multiple-lcm-of-18-and-30-factor-77726632026-03-15T22:59:14+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-number-of-clay-pots-be--7k--and-the-number-of-wooden-tools-be--4k--given-that--7k--28--solve-for--k--97921692026-04-23T09:16:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-first-six-positive-even-integers-are-20511542026-04-23T09:16:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-constraints-on-the-sequence-of-5-moves-82932322026-04-23T09:21:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-arranging-9-artifacts-where-76652592026-04-23T09:21:48+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-identify-the-prime-numbers-between-1-and-8-77684432026-04-23T09:21:49+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-x-be-the-integer-then-18049712026-04-24T19:49:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-smallest-four-digit-number-divisible-by-both-12-and-18-compute-the-least-common-multiple-lcm-of-12-and-18-29622132026-04-24T19:49:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-the-pressure-function-79609472026-04-24T19:50:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-each-hour-mia-tutors-her-net-earnings-are-53701002026-04-23T09:32:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-expanding-the-left-hand-side-16280342026-04-23T09:32:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-sequence-is-arithmetic-with-first-term--a--3--common-difference--d--4--and-last-term--l--43--the-formula-for-the--n--th-term-is--an--a--n-1d--setting--an--43--98070332026-04-23T09:47:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract--2x--from-both-sides-27512552026-04-23T09:47:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-test-divisibility-by-primes-221-is-not-divisible-by-2-3-or-5-dividing-by-13-47184432026-04-23T09:47:41+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-order-of-addition-matters-we-are-looking-for-the-number-of-permutations-of-5-catalysts-taken-2-at-a-time-this-is-given-by-80499322026-04-23T09:47:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-classic-surjection-problem-distributing-6-distinguishable-participants-to-4-distinguishable-designs-such-that-no-design-is-left-untested-the-number-of-such-surjective-functions-is-given-by-29723012026-04-23T09:47:40+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-in-a-circular-arrangement-with-alternating-roles-we-fix-one-persons-position-to-account-for-rotational-symmetry-fix-a-government-official-at-one-seat-then-the-seats-alternate-g-i-g-i-g-i-the-remaining-2-government-officials-can-be-arranged-in-2-ways-and-the-3-industry-leaders-in-3-ways-so-the-total-number-of-distinct-arrangements-is-64666552026-04-23T09:47:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-algorithm-evaluates-configurations-based-on-departure-times-and-traffic-conditions-since-the-choices-are-independent-the-total-number-of-configurations-is-the-product-of-the-number-of-departure-times-and-traffic-conditions-90301302026-04-24T19:56:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-problem-reduces-to-arranging-a-multiset-of-elements-there-are-a-total-of-2--3--1--6-transmission-events-with-2-indistinct-water-based-3-indistinct-contact-based-and-1-indistinct-airborne-method-the-number-of-distinct-sequences-is-given-by-the-multinomial-coefficient-44288902026-04-24T19:56:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-a-permutation-of-a-multiset-there-are-4--3--2--9-fish-total-with-4-indistinct-river-a-3-indistinct-river-b-and-2-indistinct-river-c-tags-the-number-of-distinct-sequences-is-48014802026-04-24T19:56:00+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-equivalent-to-arranging-a-multiset-of-4-urgent-u-3-routine-r-and-2-follow-up-f-visits-the-total-number-of-distinct-schedules-is-11931952026-04-24T19:56:03+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-angle-between-two-unit-vectors-is-given-by-the-absolute-difference-of-their-direction-angles-52154262026-03-15T23:45:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since--gx--is-a-cubic-polynomial-we-assume-74158222026-04-23T10:01:38+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-two-functions-60035892026-04-24T20:02:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--s--e1--e2---p--e1-e2--from-the-equations-52252812026-04-23T10:19:13+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-greatest-common-factor-gcf-of-36-and-48-factor-both-numbers-12755442026-04-23T10:42:26+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-analyze-the-given-equation-in-spherical-coordinates-86803352026-04-23T10:45:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--d--120---vp--6--and--vs--4--into-the-formula-65985832026-04-23T11:04:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-successful-outcomes-in-the-additional-10-hours-total-successful-outcomes--065-imes-40--x--26--x--total-hours--50--the-new-success-rate--33492032026-03-16T00:53:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--r--be-the-pre-update-retention-rate-after-the-update-retention-is--r--012--given--1000-imes-r--012--850--solve-18158342026-03-16T00:53:16+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--p---a--and--i--represent-the-cost-of-protein-algae-paste-and-insects-in-cents-the-system-is-68364942026-04-23T11:06:29+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-start-by-squaring-both-sides-of-the-given-equation-26424922026-04-23T11:10:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-quadratic-formula-98351552026-04-23T11:10:33+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-list-all-prime-numbers-less-than-30-38869622026-04-24T20:17:42+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-arithmetic-progression-be--a---2d-a---d-a-a--d-a--2d-a--3d--where--a--is-the-fourth-term-and--d--is-the-common-difference-84444042026-04-23T11:16:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-given-summation-can-be-simplified-using-partial-fraction-decomposition-82869892026-04-23T11:20:50+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-magnitude-of-the-sum-of-two-vectors-47402822026-04-23T11:23:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-total-number-of-distinct-identity-markers-we-apply-the-multiplication-principle-each-marker-consists-of-one-triangle-one-circle-and-one-square-since-the-choices-are-independent-98788642026-04-24T20:20:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-to-count-the-number-of-sequences-of-length-4-using-3-types-of-pulses-pressure-temperature-vibration-such-that-each-sequence-contains-at-least-one-of-each-type-45823852026-03-16T01:13:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-distributing-3-distinct-crafts-to-12-distinguishable-families-with-fixed-group-sizes-5-to-pottery-4-to-weaving-and-3-to-carving-96365792026-03-16T01:13:35+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-original-side-length-be--s--the-area--a--of-an-equilateral-triangle-is-given-by-the-formula-48610092026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-of-the-right-triangle-be--a--and--b--and-the-hypotenuse--z--the-area--a--of-the-triangle-is-41427682026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-be--a--9---b--12--and-hypotenuse--c--first-compute--c--50673412026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-triangle-have-legs--a--15---b--and-hypotenuse--h--the-inradius--r--of-a-right-triangle-is-84267332026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-for-an-equilateral-triangle-circumscribed-about-a-circle-ie-the-incircle-has-radius--r--6--the-relationship-between-the-side--s--and-the-inradius-is-54681352026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-diagonal-of-the-rectangle-is-the-diameter-of-the-circle-compute-the-diagonal-using-the-pythagorean-theorem-22847172026-04-24T20:21:09+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let-the-legs-be--a--5---b--and--c--13--use-pythagoras-to-find--b--9665612026-04-24T20:21:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-c-divides-ab-in-the-ratio-23-the-position-vector-of-c-is-given-by-the-section-formula-61556232026-04-24T20:21:05+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-shortest-distance-from-a-point-to-a-line-is-along-the-perpendicular-the-given-line-is-y--2x--1-so-slope-m--2-the-perpendicular-slope-is---70638102026-04-24T20:21:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-compute-this-probability-we-use-combinations-to-count-favorable-outcomes-over-total-possible-outcomes-28789002026-04-24T20:21:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-since-the-stands-are-indistinguishable-we-are-looking-for-the-number-of-unordered-partitions-of-6-distinct-objects-into-2-non-empty-unlabeled-subsets-5549252026-04-24T20:21:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-dealing-6n-cards-into-3-piles-of-2n-cards-each-38266662026-04-24T20:21:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-determine-the-number-of-experiments-we-compute-the-number-of-ways-to-choose-3-compounds-from-8-and-2-delivery-mechanisms-from-5-these-choices-are-independent-so-we-multiply-the-combinations-32651652026-04-24T20:21:18+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-substitute--t--3--into--vt--60403882026-03-16T01:18:20+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-set-the-equations-equal-at--x--2--22637162026-04-24T20:21:51+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--be-the-cost-of-a-per-liter-and--b--for-b-the-system-is-15656942026-04-23T11:27:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-of-the-triangle-can-be-computed-using-the-two-legs-34634732026-04-23T11:34:17+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-known-trigonometric-values-10798342026-04-23T11:34:52+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-we-determine-the-total-number-of-outcomes-since-each-roll-of-a-10-sided-die-has-10-possible-results-and-the-die-is-rolled-4-times-the-total-number-of-outcomes-is-84596862026-04-24T20:24:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-minimum-value-on-a-12-sided-die-is-1-and-the-maximum-is-12-we-want-the-number-of-integer-solutions-to-63835062026-04-24T20:24:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-total-number-of-ways-to-choose-4-students-from-12-9340502026-04-24T20:24:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-longest-side-of-the-triangle-is-15-and-the-altitude-drawn-to-this-side-corresponds-to-the-smallest-altitude-if-the-area-is-fixed-first-compute-the-area-using-herons-formula-the-semi-perimeter-is-82839562026-04-24T20:24:01+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-original-triangle-has-side-lengths-10-24-26-first-verify-it-is-a-right-triangle-54314702026-04-24T20:24:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-area-a-of-an-equilateral-triangle-with-side-s-is-1050012026-04-24T20:24:02+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-this-is-an-arithmetic-sequence-with-first-term-a1--3-common-difference-d--4-and-last-term-an--47-47203222026-03-16T01:29:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-total-volume-of-the-solution-since-25-of--x--is-sugar--025x--40--solve-for--x---x--87993512026-04-23T11:39:47+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-rewrite-the-line-in-standard-form-x---2y--8--0-the-distance-from-x0-y0-to-ax--by--c--0-is--280532026-04-23T11:39:45+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-using-the-pythagorean-theorem-28768762026-04-23T11:50:10+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-calculated-by-summing-all-the-numbers-and-dividing-by-the-count-of-numbers-first-we-sum-the-numbers-53969262026-04-23T11:51:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-least-common-multiple-lcm-of-12-and-15-first-determine-their-prime-factorizations-83010272026-04-23T11:51:24+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-subtract-the-amount-used-from-the-total-87084962026-04-23T11:51:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-the-arithmetic-mean-is-the-sum-divided-by-the-number-of-terms-64061992026-04-23T11:51:32+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-grams-of-type-a-then-the-grams-of-type-b-is--200---x--the-total-organic-matter-equation-is-77268282026-04-24T20:29:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--t--be-the-number-of-years-the-populations-are-30916622026-04-24T20:29:28+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--be-the-number-of-characters-in-one-instance-of-dialect-a-and--b--be-that-of-dialect-b-56333542026-03-16T01:50:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--x--be-the-nitrogen-content-per-unit-of-soil-x-and--y--be-that-of-soil-y-78491552026-03-16T03:44:11+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-let--a--be-the-length-of-a-shard-from-site-a-and--b--from-site-b-16020302026-03-16T01:50:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-multiply-both-sides-by-12-to-eliminate-denominators-3119022026-03-16T01:50:21+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-first-find-the-slope--m--1084462026-03-16T01:50:19+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-looking-for-the-least-common-multiple-lcm-of-18-and-24-29253652026-03-16T03:43:56+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-use-the-formula-for-the-magnitude-of-a-vector-23686112026-03-16T03:43:54+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-that-x-y-z--0-and-x--y--z--1-and-we-are-to-minimize-the-expression-58340402026-04-23T11:59:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-we-are-given-a-recurrence-type-functional-equation-46962242026-04-23T11:59:34+00:00daily0.8https://dev.imagemakers.co.za/guides/solution-to-find-the-intersection-we-solve-the-system-of-equations-15273082026-04-24T20:37:24+00:00daily0.8