Question

A spherical baloon is being inflated. What is the rate of increase of the surface area (S=4*pi*r^2) when r is 2ft?
Can anyone show me how to do this?

Answer 1

This is rather late, but oh well, here goes. You're looking for the rate of change of the surface area, let's call it A, which is dA/dt. So, implicitly differentiate the equation for surface area:

\[\frac{dS}{dt} = 8 \pi r \frac{dr}{dt}\]

You're given that r = 2, so the final answer is:

\[\frac{dS}{dt} = 16 \pi \frac{dr}{dt}\]

Although it doesn't quite look complete with that dr/dt in there, you can't possibly figure out the rate at which radius is increasing from the given info, so you have to leave your answer in terms of that.