Question

A spherical ball of ice melts uniformly. When the radius of the ball is $5$ $cm$, find the rate of change of its volume with respect to its radius.

Answer 1

Let the radius be $r$ and volume be $V$ of the ball at any instant, then.
$V=\frac{4}{3}\pi r^2$
Differentiating w.r.t. $r$, $\frac{dV}{dr}=\frac{4}{3}\pi \frac{d}{dr}{r}^3$
$\Rightarrow \frac{dV}{dr}=\frac{4}{3}\pi \times 3r^2$
$\Rightarrow \frac{dV}{dr}=4\pi r^2$
when $r=5$ $cm$, then $\frac{dV}{dr}=4\pi (5{)}^2$
$\frac{dV}{dr}=100\pi c{m}^3$.