Question

An open cylindrical can has to be made with $100$ square units of tin. If its volume is maximum, then the ratio of its base radius and the height is

• A. $2:1$
• B. $1:1$
• C. $1:2$
• D. $\sqrt{2}:1$

Answer 1

The correct option is B $1:1$

Let $r$ be the base radius and $h$ be the height of the cylinder.
Then, $2\pi rh+\pi r^2=100$
$\Rightarrow h=\frac{50}{\pi r}-\frac{r}{2}$
Volume of cylinder, $V=\pi r^{2}h=\pi r^2(\frac{50}{\pi r}-\frac{r}{2})=50r-\frac{\pi r^3}{2}$
$\frac{dV}{dr}=50-\frac{3\pi r^2}{2}$

$\frac{dV}{dr}=0$
$\Rightarrow r=\frac{10}{\sqrt{3\pi }}$
$\frac{d^{2}V}{d{r}^2}=-3\pi r<0$ at $r=\frac{10}{\sqrt{3\pi }}$
Hence, $V$ is maximum when $r=\frac{10}{\sqrt{3\pi }}$
$\therefore h=\frac{50}{\pi \cdot \frac{10}{\sqrt{3\pi }}}-\frac{10}{2\sqrt{3\pi }}=\frac{10}{\sqrt{3\pi }}$
So, when $V$ is maximum, $r:h=1:1$