Hello, cwarzecha!

A right circular cylinder is inscribed in a hemisphere of radius 7.

Find the radius and height of the cylinder of maximum volume.

This is a side view of the hemisphere and cylinder.

Code:

D * * * A
* - - + - - *
* | | | *
* | | | *
| | |h
* | | | *
* - + - - + - - + - *
C O r B

Draw line segment

The hemisphere has center and radius

The cylinder has the cross-section with radius and height

In right triangle .[1]

The volume of the cylinder is: . .[2]

Substitute [1] into [2]: .

And __that__ is the function you must maximize . . .