Question 11(f) | BetaMath

Question 11(f)

The volume of a sphere of radius $r$ cm, is given by $\displaystyle V = \frac43\pi r^3$ , and the volume of the sphere is increasing at a rate of $10 \text{cm}^3\text{s}^{–1}$ .

Show that the rate of increase of the radius is given by $\displaystyle \frac{dr}{dt} = \frac5{2\pi r^2}\text{cm}\text{s}^{–1}$

Solution

Recognise that the chain rule applies to relate the rates of change in volume, radius and time:

\frac{dV}{dt} = \frac{dV}{dr} \cdot \frac{dr}{dt}

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