Question

If the rate of change of volume of a sphere is equal to the rate of change of its radius, find the radius of the sphere.

Answer 1

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius and}\mathit{\text{V}}\text{be the volume of the sphere at any time}\mathit{\text{t}}\text{. Then,} \\ V=\frac{4}{3}\mathrm{\pi }{r}^3 \\ \Rightarrow \frac{dV}{dt}=4\mathrm{\pi }{r}^2\frac{dr}{dt} \\ \Rightarrow 4\mathrm{\pi }{r}^2=1\left[\because \frac{dV}{dt}=\frac{dr}{dt}\right] \\ \Rightarrow r^2=\frac{1}{4\mathrm{\pi }} \\ \Rightarrow r=\sqrt{\frac{1}{4\mathrm{\pi }}} \\ \Rightarrow r=\frac{1}{2\sqrt{\mathrm{\pi }}}\mathrm{units} \end{gathered}$$