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Home » \( v = \frac1502.5 = 60 \) kilometers per hour. - High Altitude Science

\( v = \frac1502.5 = 60 \) kilometers per hour. - High Altitude Science

📅 March 10, 2026 👤 scraface

Mar 10, 2026

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📌 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 $.

📌 \boxed{36.00}

📌 Question: In a science experiment, a drone flies along the line $ \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} 2 \\ -1 \end{pmatrix} + t \begin{pmatrix} 3 \\ 4 \end{pmatrix} $. Find the point on this line closest to $ (5, 7) $.

📌 Solution: The closest point occurs where the vector from $ (5, 7) $ to the line is perpendicular to the direction vector $ \begin{pmatrix} 3 \\ 4 \end{pmatrix} $. Let the point on the line be $ (2 + 3t, -1 + 4t) $. The vector from $ (5, 7) $ to this point is $ \begin{pmatrix} 2 + 3t - 5 \\ -1 + 4t - 7 \end{pmatrix} = \begin{pmatrix} -3 + 3t \\ -8 + 4t \end{pmatrix} $. Dot product with $ \begin{pmatrix} 3 \\ 4 \end{pmatrix} $ must be zero: $ 3(-3 + 3t) + 4(-8 + 4t) = 0 $. Simplify: $ -9 + 9t - 32 + 16t = 0 \Rightarrow 25t - 41 = 0 \Rightarrow t = \frac{41}{25} $. Substitute back: $ x = 2 + 3 \cdot \frac{41}{25} = \frac{173}{25} $, $ y = -1 + 4 \cdot \frac{41}{25} = \frac{156}{25} $.

📌 \boxed{\left( \dfrac{173}{25}, \dfrac{156}{25} \right)}

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