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Question
Find the surface area of a sphere when its volume is changing at the same rate as its radius ?
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Solution
\[\text{Let r be the radius and V be the volume of the sphere at any time t . Then }, \]
\[V = \frac{4}{3}\pi r^3 \]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \left( \frac{dr}{dt} \right)\]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \left( \frac{dV}{dt} \right) \left[ \because \frac{dV}{dt} = \frac{dr}{dt} \right]\]
\[ \Rightarrow 4\pi r^2 = 1 \]
\[ \Rightarrow \text { Surface area of sphere =1 square unit }\]
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