Question

Let the radius and the height of a right circular cylinder is related as $r^2+h=6$. If the volume of the cylinder is maximum, then $\frac{r}{h}$ is -

• A. $\frac{1}{\sqrt{3}}$
• B. $\sqrt{3}$
• C. $1$
• D. $3$

Answer 1

The correct option is A $\frac{1}{\sqrt{3}}$

$V=\pi r^{2}h$
$\Rightarrow V=\pi r^2(6-r^2)$
$\Rightarrow \frac{dV}{dr}=\pi (12r-4r^3)$
For maximum or minimum,
$\frac{dV}{dr}=0$
$\Rightarrow r^2=3\Rightarrow h=3$

$\Rightarrow \frac{r}{h}=\frac{1}{\sqrt{3}}$