Question

A balloon which remains spherical has a variable radius. Find the rate at which its volume is increasing with the radius when the radius is 10 cm.

Answer 1

Let r be the radius of spherical balloon and V be its volume.
Then, $r=10cm$ and $V=\frac{4}{3}\pi r^3$,
Rate of change of volume w.r.t radius r, $\frac{dV}{dr}=\left(\frac{4}{3}\pi \right)3r^2$
(differentiating w.r.t. t)
$=4\pi r^2=4\pi (10{)}^2=400\pi (\because r=10cm)$
Hence, the volume of the balloon is increasing at the rate of $400\pi c{m}^3/cm$.