Hello,

1. make a sketch: see attachment

2. Volume of the cylinder:

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

. Since R = 1 you'll get

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

You'll get the extreme (minimum or maximum) value of if

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

EDIT: Forgot to attach the sketch. Sorry!