Let’s begin with the formula for volume and surface area: V = (4/3) π r3 and S = 4 π r2. You’re going to want to take the derivative of those two with respect to time. Note that by the chain rule, a function of r given by f(r) has the derivative (df/dr) (dr/dt). You should find that dV/dt = 4 π r2 dr/dt and dS/dt = 8 π r dr/dt. Now simply compare their formulas to identify whatever relationships you’d like!


Several ways. From the way you wrote the question, I’m assuming you don’t want your answer to depend on dr/dt, so solve both of those equations for dr/dt and then set them equal to each other. This will give you a direct relationship, but it’s up to you what to do with it. You can solve for one in terms of the other, find their proportion, or whatever else.

S = dV/dr

So if you take d/dt on both sides then...?