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: $\frac{dV}{dt}$. Using the chain rule, this is just:

$\frac{dV}{dt} = \frac{dV}{dr} \times \frac{dr}{dt}.$

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:

$\frac{dV}{dr} = 4\pi r^2.$

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

$\frac{dr}{dt} = 2t.$

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

$\frac{dV}{dt} = 4\pi r^2 \times 2t = 8\pi r^2 t.$

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

$\frac{dV}{dt} = 4\pi t(t^2 + 2)^2.$