show that the radius of right circular cylinder of maximum volume that can be inacribed in a sphere of radius 18cm is 8√6

I assume you made a sketch.

I sketched a right-angled triangle by letting the radius of the circular base of the cylinder to be r and the distance to the centre of the sphere to be be h
then h^2 + r^2 = 18^2 ----> r^2 = 324 - h^2

volume of cylinder
= π r^2 (2h)
= π(324 - h^2)(2h)
=648πh - 2πh^3
d(volume)/dh = 648π - 6πh^2
= 0 for a max of volume

648π - 6πh^2 = 0
divide out 6π
108 - h^2 = 0
h^2 = 108
so r^2 = 324 - 108 = 216
r = √216 = 6√6

You said it should be 8√6
Check my arithmetic, I can't seem to find an error