At time t seconds the radius of a sphere is r cm, its volume is V cm³ and its surface area is S cm² - Edexcel - A-Level Maths Pure - Question 8 - 2014 - Paper 8

To find ( \frac{dr}{dt} ), we start by differentiating the volume equation with respect to time:

1. The formula for the volume of a sphere is given by: $V = \frac{4}{3} \pi r^3$

2. Differentiate both sides with respect to time t: $\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}$

3. We know ( \frac{dV}{dt} = 3 ) cm³/s, thus substituting this into the equation gives: $3 = 4 \pi r^2 \frac{dr}{dt}$

4. Substitute ( r = 4 ) cm into the equation: $3 = 4 \pi (4^2) \frac{dr}{dt}$ $3 = 4 \pi (16) \frac{dr}{dt}$ $3 = 64 \pi \frac{dr}{dt}$

5. Rearranging gives: $\frac{dr}{dt} = \frac{3}{64 \pi}$

6. Calculate the value: $\frac{dr}{dt} \approx 0.01492 \text{ (cm/s)}$ ( \frac{dr}{dt} \approx 0.015 \text{ (to 3 significant figures)} )