Question

Find the rate of change of the volume of a ball with respect to its radius r. How fast is the volume changing with respect to the radius when the radius is 2 cm?

Answer 1

Let V be the volume of the spherical ball. Then,
V = $\frac{4}{3}\mathrm{\pi }{r}^3$

$\Rightarrow \frac{dV}{dr}=4\mathrm{\pi }{r}^2$
Thus, the rate of change of the volume of the sphere is $4\mathrm{\pi }{r}^2$.

$$\begin{gathered} \mathrm{When}r=2\mathrm{cm}, \\ {\left(\frac{dV}{dr}\right)}_{r=2}=4\mathrm{\pi }{{\left(2\right)}^2}_{} \\ =16\mathrm{\pi }{\mathrm{cm}}^3/\mathrm{cm} \end{gathered}$$