Question

If the rate of change of volume of a sphere is equal to the rate of change of its radius, then its radius is equal to

Options

• 1 unit

• \[\sqrt{2\pi} \text { units }\]

• \[\frac{1}{\sqrt{2\pi}} \text { unit }\]

• \[\frac{1}{2\sqrt{\pi}} \text { unit}\]

Answer 1

\[\frac{1}{2\sqrt{\pi}} \text { unit }\]

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

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

\[\Rightarrow\frac{dV}{dt}=\frac{4}{3}\left( 3\pi r^2 \right)\frac{dr}{dt}\]

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

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

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

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

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