Question

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

Answer 1

The correct answer is $400π cm^3 /s.$
The volume of a sphere $(V)$ with radius $(r)$ is given by $v=\frac{4}{3}πr^3$.
Rate of change of volume $(V)$ with respect to its radius $(r)$ is given by,
$\frac{dv}{dr}=\frac{d}{dr}(\frac{4}{3}πr^3)=\frac{4}{3}π(3r^2)=4πr^2$
Therefore, when radius=10 cm,
$\frac{dv}{dr}=4π(10)^2=400π$
Hence, the volume of the balloon is increasing at the rate of $400π cm^3 /s.$