Question

Find the surface area of a sphere when its volume is changing at the same rate as its radius.

Answer 1

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