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

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

$$V=\frac{4}{3}\pi r^3$$

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

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

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

$$\Rightarrow r=\sqrt{\frac{1}{4\pi }}$$

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