A sphere radius that intersects the edge of the cylinder base is the hypotenuse of a right triangle with legs that are cylinder radius and one-half of the cylinder height. Given a sphere with radius 6, the radius and height of the cylinder are related thus:
The volume of the cylinder is:
Substituting:
Simplifying:
Take the first derivative
Set the first derivative equal to zero and solve for the value of the height that gives the maximum volume.
John
My calculator said it, I believe it, that settles it
