Question

You are blowing air into a spherical balloon at a rate of 7 cubic inches per second. The goal of this problem is to answer the following question: What is the rate of change of the surface area of the balloon at time t= 1 second, given that the balloon has a radius of 3 inches at that instant?

(a) Next write a formula relating the changing volume V(t) of the sphere to the changing radius r(t), and differentiate that formula with respect to t. Using what you know about V'(t) and r(1), find the rate of change of the radius at t=1 sec:

(b) Finally, write a formula relating the changing surface area S(t) of the sphere to the changing radius r(t), and differentiate that formula with respect to t. Use what you know about r(1) and r'(1) to determine the rate of change of the surface area at t=1 sec:

Answer 1

a = 4πr^2
da/dt = 8πr dr/dt