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प्रश्न
If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere ?
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उत्तर
\[\text { Let r be the radius and V be the volume of the sphere at any time t. Then },\]
\[V = \frac{4}{3}\pi r^3 \]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\]
\[ \Rightarrow 4\pi r^2 = 1 \left[ \because \frac{dV}{dt} = \frac{dr}{dt} \right]\]
\[ \Rightarrow r^2 = \frac{1}{4\pi}\]
\[ \Rightarrow r = \sqrt{\frac{1}{4\pi}}\]
\[ \Rightarrow r = \frac{1}{2\sqrt{\pi}} \text { units }\]
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