Question

The radius of an air bubble is increasing at the rate of $0.5$ centimeter per second. At what rate is the volume of the bubble increasing when the radius is $1$ centimeter ?

Answer 1

The radius of a bubble is increasing at the Rate $0.5cm/sec$.

$\frac{dr}{dt}=0.5cm/sec.$

Rate of inc. of the volume

$v=\frac{4}{3}\pi r^3$

$\frac{dv}{dr}=\frac{4}{3}\pi 3r^2\frac{dr}{dt}$

$=\frac{4}{3}\times \pi \times 3\times (1{)}^2\times 0.5$

$\frac{dv}{dr}=2\pi c{m}^3/sec$

This is the rate of increase in the volume of the bubble.