Question

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

Answer 1

Let $r$ be the radius of the ballon and $V$ be its volume at any time $t.$

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

Differentiate both sides w.r.t. $t,$ we get

$\Rightarrow$ $\frac{dV}{dt}=\frac{4}{3}\pi .3r^2\frac{dr}{dt}$

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

It is given that $\frac{dV}{dt}=900c{m}^2/s$

$\Rightarrow$ $900=4\pi r^2\frac{dr}{dt}$

$\Rightarrow$ $\frac{dr}{dt}=\frac{225}{\pi r^2}$

When, $r=15cm$

$\Rightarrow$ $\frac{dr}{dt}=\frac{225}{\pi \times (15{)}^2}$

$\Rightarrow$ $\frac{dr}{dt}=\frac{1}{\pi }cm/s$

$\therefore$ Radius of the balloon is increasing at the rate of $\frac{1}{\pi }cm/s$ when the radius is $15cm.$