Question

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 ?

Answer 1

\[\text { Let r be the radius and V be the volume of the sphere at any time t. Then },\]

\[V = \frac{4}{3}\pi r^3 \]

\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}\]

\[ \Rightarrow 4\pi r^2 = 1 \left[ \because \frac{dV}{dt} = \frac{dr}{dt} \right]\]

\[ \Rightarrow r^2 = \frac{1}{4\pi}\]

\[ \Rightarrow r = \sqrt{\frac{1}{4\pi}}\]

\[ \Rightarrow r = \frac{1}{2\sqrt{\pi}} \text { units }\]