Question

A cylinder is inscribed in a sphere of radius r. What is the height of the cylinder of maximum volume?

• A. $\frac{2r}{\sqrt{3}}$
• B. $\frac{r}{\sqrt{3}}$
• C. $2r$
• D. $\sqrt{3}$r

Answer 1

The correct option is A $\frac{2r}{\sqrt{3}}$

We know that volume of the cylinder $V=\pi (radius{)}^2\times (height)$

here radius is r then

$\Rightarrow \pi r^2-\frac{3}{4}\times \pi h^2=0$

$\Rightarrow \frac{3}{4}\times \pi h^2=\pi r^2$

$\Rightarrow h^2=\frac{\pi r^2}{\frac{3}{4}\pi }$

$\Rightarrow h=\sqrt{\frac{4r^2}{3}}$

$\Rightarrow h=\frac{2r}{\sqrt{3}}$.