Question

The radius of an air bubble is increasing at the rate of $0.5$ cm/sec. At what rate is the volume of the bubble increasing when the radius is $1$ cm?

Answer 1

We have, $V=\frac{4}{3}\pi r^3$ and $\frac{dr}{dt}=0.5$
$V=\frac{4}{3}\pi r^3\Rightarrow \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\Rightarrow \frac{dV}{dt}=4\pi \left(1{)}^2\right(0.5)=2\pi$