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Home » To find $\cos \theta$, we use the formula for the dot product: - Imagemakers

To find $\cos \theta$, we use the formula for the dot product: - Imagemakers

📅 May 2, 2026 👤 scraface

May 02, 2026

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📌 Now, the ratio of the area of the circle to the area of the triangle is:

📌 \text{Ratio} = \frac{A_{\text{circle}}}{A_{\text{triangle}}} = \frac{\frac{\pi s^2}{12}}{\frac{\sqrt{3}}{4} s^2} = \frac{\pi s^2}{12} \cdot \frac{4}{\sqrt{3} s^2} = \frac{\pi}{3\sqrt{3}} = \frac{\pi \sqrt{3}}{9}

📌 Thus, the ratio of the area of the circle to the area of the triangle is \(\boxed{\frac{\pi \sqrt{3}}{9}}\).

📌 A regular tetrahedron has a side length of \( b \) units. What is the ratio of the volume of the inscribed sphere to the volume of the tetrahedron?

📌 The volume \( V_{\text{tetrahedron}} \) of a regular tetrahedron with side length \( b \) is:

📌 V_{\text{tetrahedron}} = \frac{b^3}{6\sqrt{2}}

📌 The radius \( r \) of the inscribed sphere in a regular tetrahedron is:

📌 r = \frac{b \sqrt{6}}{12}

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4! \times 3! = 24 \times 6 = \boxed{144}
Pregunta: Calcula \(\dbinom{12}{5} - \dbinom{12}{4}\).
Solución: Usando las propiedades de los coeficientes binomiales:
\dbinom{12}{5} = \dbinom{12}{7}, \quad \dbinom{12}{4} = \dbinom{12}{8}
Pero directamente:
\dbinom{12}{5} = 792, \quad \dbinom{12}{4} = 495
Así:
- 495 = \boxed{297}

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