Prove that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is Also find the maximum volume.

The radius of cylinder = r

The radius of sphere = R

The height of cylinder = H

Now acc. to the diagram:

r = R cosθ …(1)

…(2)

Now volume of cylinder =

For V to be maximum

…(3)

…(4)

Put (4) in (2),

We get:

And

Maximum Volume of cylinder = πr² H