Question

Two measurements of a cylinder are varying in such a way that the volume is kept constant. If the rates of change of the radius (r) and height (h) are equal in magnitude but the opposite signs then

• A. $r=2h$
• B. $h=2r$
• C. $h=r$
• D. $h=4r$

Answer 1

The correct option is A

$r=2h$

Find the relation

Given,

$\frac{dr}{dt}=-\frac{dh}{dt}$

Volume of cylinder $=V={\mathrm{\pi r}}^2\mathrm{h}$

Now, volume is constant, $\frac{dV}{dt}=0$

$$\begin{gathered} \Rightarrow \mathrm{\pi h}2\mathrm{r}\frac{dr}{dt}+{\mathrm{\pi r}}^2\frac{d\mathrm{h}}{d\mathrm{t}}=0 \\ \Rightarrow \mathrm{\pi h}2\mathrm{r}\frac{d\mathrm{r}}{d\mathrm{t}}-{\mathrm{\pi r}}^2\frac{d\mathrm{r}}{d\mathrm{t}}=0 \\ \Rightarrow \mathrm{r}=2\mathrm{h} \end{gathered}$$

Hence, the correct option is $A$.