The volume of a spherical bubble is increasing at a constant rate - AQA - A-Level Maths Pure - Question 10 - 2019 - Paper 1

Since we know the volume is increasing at a constant rate, we denote this rate as $k$:

$\frac{dV}{dt} = k$

Substituting this into our earlier equation gives:

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

Rearranging the equation to isolate rac{dr}{dt} yields:

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

From the equation, it is clear that the rate of increase of the radius rac{dr}{dt} is inversely proportional to $r^2$:

$\frac{dr}{dt} \propto \frac{1}{r^2}$

Thus, we have demonstrated that the rate of increase of the radius of the bubble is inversely proportional to the square of the radius.