I need to solve the following problem applying the chain rule.
The volume $V$ of a sphere of radius $r$ is given by the formula
$$V=\frac{4\pi r^3}{3}.$$
Suppose that a snowball initially of radius $r=100$ cm is shrinking uniformly at the rate of $1$ cm per hour, so that
$$r=r(t)=100-t.$$
Use the Chain Rule to find a formula for $\frac{dV}{dt}$, the rate at which the volume $V$ of the snowball is shrinking (in cubic centimetres per hour) after $t$ hours have elapsed.
I know that, using the chain rule, $\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}$ but I can't really understand how to solve this problem.
Best Answer
Yeah, as you know, apply chain rule and replace $r$ by $100-t$.
So, $$\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} = 4\pi(100-t)^2(-1) = -4\pi(100-t)^2$$
As $\frac{dV}{dt} $ is negative, the volume is clearly decreasing.