Question

If at any instant $t$, for a sphere, $r$denotes the radius, $S$ denotes the surface area and $V$ denotes the volume, then what is $dV/dt$equal to?

• A. $\frac{1}{2}S\frac{dr}{dt}$
• B. $\frac{1}{2}r\frac{dS}{dt}$
• C. $r\frac{dS}{dt}$
• D. $\frac{1}{2}{r}^2\frac{dS}{dt}$

Answer 1

The correct option is B

$\frac{1}{2}r\frac{dS}{dt}$

Finding the value of :

The surface area of a sphere $S=4\pi r^2$

Differentiate both sides with respect to $t$

$$\begin{gathered} \frac{dS}{dt}=8\pi r\frac{dr}{dt} \\ \Rightarrow \frac{dr}{dt}=\frac{\frac{dS}{dt}}{8\pi r} \end{gathered}$$

$$\begin{gathered} \Rightarrow \frac{dV}{dt}=4\pi r^2\left(\frac{dr}{dt}\right) \\ \Rightarrow \frac{dV}{dt}=4\pi r^2\left(\frac{{\displaystyle \frac{dS}{dt}}}{8\pi r}\right) \\ \Rightarrow \frac{dV}{dt}=\frac{r}{2}\left(\frac{dS}{dt}\right) \end{gathered}$$$\mathit{V}\mathbf{}\mathbf{=}\mathbf{}\mathbf{(}\mathbf{4}\mathbf{/}\mathbf{3}\mathbf{)}\mathit{\pi }{\mathit{r}}^{\mathbf{3}}$

Hence the correct option is B.