Question

A closed cylinder with a given volume $V$ and minimum surface area has height equal to $k$ times the radius. Then value of $k$ is

Answer 1

Let $H$ be the height, $R$ be the base radius, $V$ be the volume (fixed) and $S$ be the total surface area (to be minimum) of the cylinder.
$V=\pi R^{2}H$
$\Rightarrow H=\frac{V}{\pi R^2}\cdots \left(i\right)$
$S=2\pi R^2+2\pi RH$
$\Rightarrow S=2\pi R^2+\frac{2V}{R}$
Differentiating w.r.t. $R,$
$S^{\prime }=4\pi R-\frac{2V}{R^2}$
and $S^{\prime \prime }=4\pi +\frac{4V}{R^3}>0$
$\therefore S$ is minimum when $S^{\prime }=0$
$\Rightarrow V=2\pi R^3$
$\Rightarrow H=\frac{2\pi R^3}{\pi R^2}($From $\left(i\right))$
$\Rightarrow H=2R$