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Home » Say Goodbye to Mess: The Hair Wrap That’s Taking Over Contour Aesthetics! - Abbey Badges

Say Goodbye to Mess: The Hair Wrap That’s Taking Over Contour Aesthetics! - Abbey Badges

📅 February 27, 2026 👤 scraface

Feb 27, 2026

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📌 Solution: The volume of a sphere is given by $ V = \frac{4}{3} \pi r^3 $. Differentiating with respect to time $ t $, we get:

📌 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}} $.

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