Question

A balloon which always remains spherical has a variable radius. The rate at which its volume is increasing is $m\pi$ $c{m}^2$ with the radius when the later is $10$ cm. Find $m$

Answer 1

The volume of a sphere $\left(V\right)$ with radius $\left(r\right)$ is given by $V=\frac{4}{3}\pi r^3$.
Rate of change of volume $\left(V\right)$ with respect to its radius $\left(r\right)$ is given by,
$\frac{dV}{dr}=\frac{d}{dr}\left(\frac{4}{3}\pi r^3\right)=\frac{4}{3}\pi \left(3\pi r^2\right)=4\pi r^2$
Therefore, when radius $=10cm$,
$\frac{dV}{dr}=4\pi (10{)}^2=400\pi c{m}^2$