Question

A balloon which always remains spherical is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.

Answer 1

Given: Spherical balloon inflated by pumping in 900 cubic centimetres of gas per second

To find the rate at which the radius of the balloon is increasing when the radius is 15 cm

Let the radius of the given spherical balloon be r cm and let V be the volume of the spherical balloon at any instant time

Then according to the given criteria,

As the balloon is inflated by pumping 900 cubic centimetres of gas per second hence the rate of volume of the spherical balloon increases by, \(\frac{dV}{dt}\) = 900 cm³/sec ...(i)

We know volume of the spherical balloon is V = \(\frac{4}{3}\pi r^3.\)

Applying derivative with respect to time on both sides we get,

So when the radius is 15cm, the above equation becomes,

Hence the rate at which the radius of the balloon is increasing when the radius is 15 cm will be \(\frac{1}{\pi}\text{cm/sec.}\)