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📅 May 3, 2026 👤 scraface

May 03, 2026

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📌 5Question: An AI-powered medical imaging system processes a signal modeled by the complex number $ z = \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) $. Compute $ z^6 $ and express it in the form $ a + bi $.

📌 Solution: Using De Moivre's Theorem, $ z^n = \cos\left(n\theta\right) + i \sin\left(n\theta\right) $. Here, $ z = \cos\left(\frac{\pi}{6}\right) + i \sin\left(\frac{\pi}{6}\right) $, so $ z^6 = \cos\left(6 \cdot \frac{\pi}{6}\right) + i \sin\left(6 \cdot \frac{\pi}{6}\right) = \cos(\pi) + i \sin(\pi) $. Evaluating, $ \cos(\pi) = -1 $ and $ \sin(\pi) = 0 $, so $ z^6 = -1 + 0i $.

📌 \boxed{-1}

📌 Question: A pharmacologist models the decay of a drug's effectiveness with $ D(t) = D_0 \cdot 2^{-t/12} $. Determine the time $ t $ when the remaining dose is $ \frac{1}{8}D_0 $, rounded to two decimal places.

📌 Solution: Set $ \frac{1}{8}D_0 = D_0 \cdot 2^{-t/12} $. Divide both sides by $ D_0 $: $ \frac{1}{8} = 2^{-t/12} $. Note $ \frac{1}{8} = 2^{-3} $, so $ 2^{-3} = 2^{-t/12} $. Equate exponents: $ -3 = -\frac{t}{12} $. Solve for $ t $: $ t = 36 $.

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