Originally Posted by

**earboth** Hello,

1. make a sketch: see attachment

2. Volume of the cylinder: $\displaystyle V_{cyl}=\pi r^2 \cdot h$

3. Relation between the height h, the radius r of the base of the cylinder and the radius R of the sphere:

$\displaystyle R^2=r^2+\frac14 \cdot h^2$ . Since R = 1 you'll get $\displaystyle r^2 = 1-\frac14 \cdot h^2$

Plug in this term into the equation to calculate the volume of the cylinder:

$\displaystyle V_{cyl}(h)=\pi \left( 1-\frac14 \cdot h^2 \right) \cdot h=\pi h - \frac14 \pi h^3$

You'll get the extreme (minimum or maximum) value of $\displaystyle V_{cyl}(h)$ if $\displaystyle V_{cyl}'(h) = 0$

$\displaystyle V_{cyl}'(h)=\pi - \frac34 \pi h^2$

$\displaystyle \pi - \frac34 \pi h^2 = 0~\implies~h=\frac23 \cdot \sqrt{3}$

Plug in this value into the equation to calculate rē. You'll get:

$\displaystyle r^2=1-\frac14 \cdot \left( \frac23 \cdot \sqrt{3} \right)^2 ~\implies~r= \frac13 \cdot \sqrt{6}$