A spherical balloon is being inflated. Find the rate of change of the surface area of the balloon with respect to the radius when the radius is 10cm.

Question

I understand that we are supposed to use derivatives in solving this question. However, I am unsure as to the process that I should follow in order to solve this question. I would appreciate an explanation of the steps to get the correct answer, thanks so much!

Answer 1

First, you need an equation for the surface area of a sphere.  That would be A=4πr²

 

For what is referred to as "related rates", you need an equation that has both of the rates you are interested in.  For this problem, we want the rate for the surface area, and hmmm, I was expecting to know the rate of change of the radius.  Were you given that info?  Without that, we really can't finish the problem.  But here are the typical steps:

 

• A=4πr² (equation that relates our variables)
  
• dA/dt = 8πr dr/dt (take the derivative of both sides with respect to time)
• dA/dt = 80π dr/dt (Plug in what you know, and again, I would expect to be given dr/dt in cm/second)