Asphere is expanding with time. the volume of the sphere is v=4pi*r^3/3, r(t)=t^2+2. what is the formula for the rate of change of the volume of the balloon, with respect to time?

We aim to find: . Using the chain rule, this is just:

Now, the first part we need to find is relatively straight forward. We are given V in terms of r, so using the power rule, this gives us:

We're also given r in terms of t, so simply differentiating r wrt t gives us:

Putting all of this together gives us our final piece of the puzzle:

But we don't want the rate of change to be in "r" and "t". So letting gives us:

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step-by-step explanation: