Find the surface area of a sphere when its volume is changing at the same rate as its radius.

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

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

$\Rightarrow \frac{d V}{d t}=4 \pi r^{2}\left(\frac{d r}{d t}\right)$

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

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

$\Rightarrow$ Surface area of sphere $=1$ square unit