A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15$ cm.

Let us assume that at any instant of time $t$, the radius of the balloon is $r$ and its volume be $V$, then
$V = \cfrac{4}{3}\pi {r^3}$ …(i)

Differentiating (i) w.r.t. $t$, we get
$\cfrac{{dV}}{{dt}} = \left( {\cfrac{4}{3}\pi } \right)\left( {3{r^2}\cfrac{{dr}}{{dt}}} \right)$

$\Rightarrow 900{\rm{c}}{{\rm{m}}^{\rm{3}}}{\rm{/sec}} = \left( {\cfrac{4}{3}\pi } \right)\left\{ {3{{(15cm)}^2}\cfrac{{dr}}{{dt}}} \right\}$

$\Rightarrow \cfrac{{dr}}{{dt}} = \cfrac{{900}}{{4\pi \times {{\left( {15} \right)}^2}}}{\rm{cm/sec}} = \cfrac{1}{\pi }{\rm{cm/sec}}$

Therefore, Rate of increase of the radius of the balloon $= \cfrac{1}{\pi }{\rm{cm/sec}}$.