Mathematics - Application of Derivative Question with Solution | TestHub

$$\begin{array}{rl}& \mathrm{V}=\frac{4}{3}\pi r^3\Rightarrow \frac{d\mathrm{V}}{dt}=4\pi r^2\cdot \frac{dr}{dt}\\ & \mathrm{S}=4\pi r^2\Rightarrow \frac{d\mathrm{S}}{dt}=8\pi r\cdot \frac{dr}{dt}\\ & \Rightarrow 8=8\pi r\frac{dr}{dt}\Rightarrow \frac{dr}{dt}=\frac{1}{\pi r}\end{array}$$ Putting the value of $\frac{dr}{dt}$ in (i), we get $$\frac{d\mathrm{V}}{dt}=4\pi r^2\times \frac{1}{\pi r}=4r$$ $\Rightarrow \frac{d\mathrm{V}}{dt}$ is proportional to $r$.