Question

The radius of a balloon which is always spherical, is variable. Find the rate of change of its volume with respect to radius when the radius is $10$ cm.

Answer 1

Volume of the sphere is $v=\frac{4}{3}\pi r^3$

Radius $r=10$ cm

Differentiating with respect to $r$ on both sides of $v$, we get

$\frac{dv}{dr}=\frac{4}{3}\pi \left(3r^2\right)$

Substitute $r=10$, we get

$\frac{dv}{dr}=\frac{4}{3}\pi (3\times {10}^2)$

$\frac{dv}{dr}=400\pi {\text{cm}}^3$

Hence the rate of change of its valume with respect to the radius is $400\pi {\text{cm}}^3/\text{sec}$