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Home » \( b^2 = 9 \Rightarrow b = 3 \) - AMAZONAWS

\( b^2 = 9 \Rightarrow b = 3 \) - AMAZONAWS

📅 March 5, 2026 👤 scraface

Mar 05, 2026

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📌 We are given that $ \frac{dV}{dt} = k \sin \theta $, and $ \theta = 60^\circ $, so $ \sin \theta = \frac{\sqrt{3}}{2} $. Substituting:

📌 k \cdot \frac{\sqrt{3}}{2} = 4\pi r^2 \frac{dr}{dt}

📌 Solving for $ \frac{dr}{dt} $:

📌 \frac{dr}{dt} = \frac{k \sqrt{3}}{8\pi r^2}

📌 The rate of volume increase is given directly as $ \frac{dV}{dt} = k \cdot \frac{\sqrt{3}}{2} $, so the rate is:

📌 \frac{dV}{dt} = \frac{k\sqrt{3}}{2}

📌 Thus, the rate of volume increase at $ \theta = 60^\circ $ is $ \boxed{\frac{k\sqrt{3}}{2}} $.

📌 Question: A data analyst plots a triangle on a coordinate plane with side lengths corresponding to data intervals of 13 cm, 14 cm, and 15 cm. What is the length of the longest altitude of this triangle?

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