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.

Let r be the radius and V be the volume of the sphere at any time t. Then,

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

$\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$

$\Rightarrow 4 \pi r^{2}=1 \quad\left[\because \frac{d V}{d t}=\frac{d r}{d t}\right]$

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

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

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