Question

If $h$ is the height and $r$ is the radius of the base of a right circular cylinder of greatest volume that can be inscribed in a given sphere, then $h$ is equal to

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

Answer 1

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

Note that $h$ refers to half of the total height of the cylinder.

To find volume of cylinder we need to multiply the area of the top by total height of cylinder.

$v=\pi (\text{radius of cylinder}{)}^2\times \left(\text{height of cylinder}\right)$

$=\pi {\left(\sqrt{r^2-h^2}\right)}^2\times 2h$

$=2\pi h(r^2-h^2)$

Now we take the derivative of the volume function and set it equal to $zero.$

If we move $h$ inside parenthesis, we need to only use power rule to set derivative.

$h=2\pi (r^{2}h-h^3)$

$\frac{dv}{dx}h=2\pi (r^2-3h^2)=0$

The $2\pi$ divides out what we are left with,

$r^2-3h^2=0⟹h^2=\frac{r^2}{3}$

$\therefore h=\frac{r}{\sqrt{3}}$

$v=2\pi h(r^2-h^2)=2\pi \frac{r}{\sqrt{3}}\{r^2-(\frac{r}{\sqrt{3}}{)}^2\}=\frac{2\pi r}{\sqrt{3}}(\frac{2r^2}{3})=\frac{4\sqrt{3}\pi r^3}{9}$