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.

## firstdwarf

Let’s begin with the formula for volume and surface area: V = (4/3) π r

^{3}and S = 4 π r^{2.}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 r^{2}dr/dt and dS/dt = 8 π r dr/dt. Now simply compare their formulas to identify whatever relationships you’d like!## [deleted]

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## firstdwarf

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.

## jdorje

S = dV/dr

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