Question

Air is being pumped into a balloon in the shape of a sphere so that its volume is increasing at a constantrate of 50^3s^-1. Find the rate at which the radius of the balloon is increasing when the radius is 10.

Answer 1

V=4πr³/3 is the volume V of a sphere of radius r.

It’s not clear from the question what the units are, but let’s assume the volume is in cm³, then the radius will be in cm.

So dV/dt=4πr²dr/dt where dV/dt=50 cm³/s is the rate of change of the volume and dr/dt that of the radius in cm/s.

Therefore, 4πr²dr/dt=50.

If the initial radius is 10cm, then 400πdr/dt=50 and dr/dt=1/(8π) cm/s. If π=3.142 the rate of change of the radius is about 0.04cm/s.