Question

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 $15cm$.

Answer 1

The volume of a sphere $\left(V\right)$ with radius $\left(r\right)$ is given by,
$V=\frac{4}{3}\pi r^3$

$\therefore$ Rate of change of volume $\left(V\right)$ with respect to time $\left(t\right)$ is given by,

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

It is given that $\frac{dV}{dt}=900c{m}^3/s$.

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

$\Rightarrow \frac{dr}{dt}=\frac{900}{4\pi r^2}=\frac{225}{\pi r^2}$

Therefore, when radius $=15$ cm,
$\frac{dr}{dt}=\frac{225}{\pi (15{)}^2}=\frac{1}{\pi }$ cm/s