Question

A sphere 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?

Answer 1

We aim to find:
`(dV)/(dt)`. Using the chain rule, this is just:

`(dV)/(dt) = (dV)/(dr) * (dr)/(dt).`

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:

`(dV)/(dr) = 4\pi r^2.`

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

`(dr)/(dt) = 2t.`

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

`(dV)/(dt) = 4\pi r^2 * 2t = 8\pi r^2 t.`

But we don't want the rate of change to be in "r" and "t". So letting
`r = t^2 + 2` gives us:

`(dV)/(dt) = 4\pi t(t^2 + 2)^2.`