Air is being pumped into a spherical balloon so that its volume increases at a rate of 70 cm3/s. How fast is the surface area of the balloon increasing when its radius is 5 cm3?
A ball with radius r has the following volume (V) and surface area (S):
V = (4/3)(πr3)
S = 4πr2
The Attempt at a Solution
The problem seems to be asking for dS/dt at r = 5. dV/dt appears to be equivalent to S.
dS/dt = (d/dt)(4πr2)
dS/dt = 8πr(dV/dt) <--- This step is the one that's really bugging me, especially that dV/dt tagged onto the end.
dS/dt = 8π5(70)
dS/dt = 2800π
dS/dt = 8796.459
Which is apparently incorrect. I'm having a lot of trouble grasping these and other related rate problems, mainly because I am not sure how to properly set up the equations and every tutorial I've found on the subject (including my textbook) simply breeze over that step by saying "Find an equation that describes the given relationship."
Any help is much appreciated.