Question

A cylinder such that the sum of its height and circumference of its base is $10\text{m}$. Find the greatest volume of the cylinder?

Answer 1

Let $h$ be the height and $r$ be the radius of the circular base

Now, as per the question,

$h+2\pi r=10\Rightarrow h=(10-2\pi r)$

Volume of cylinder $\left(V\right)=\pi r^{2}h$

$V=\pi r^2(10-2\pi r)$

$\frac{d\left(V\right)}{dr}=0$ ($\because$ for max or min)

$0=10\pi \times 2r-2{\pi }^2\times 3r^2$

$2\pi r(10-3\pi r)=0$

$r=\frac{10}{3\pi }$

Now, $\frac{d^2\left(V\right)}{d{r}^2}=20\pi -12{\pi }^{2}r=20\pi -12\times {\pi }^2\times \frac{10}{3\pi }$

$=-20\pi <0$ (max at $r=\frac{10}{3\pi })$

Max. Volume $=\pi {\left(\frac{10}{3\pi }\right)}^2[10-2\pi \times \frac{10}{3\pi }]$

$=\left[\frac{1000}{27\pi }\right]{\text{m}}^3$