Question

Show that the right circular cylinder of given surface and maximum volume is such that is heights is equal to the diameter of the base.

Answer 1

Let r and h be the radius and height of the cylinder respectively.

Then, the surface area (S) of the cylinder is given by,

Let V be the volume of the cylinder. Then,

∴ By second derivative test, the volume is the maximum when.

Hence, the volume is the maximum when the height is twice the radius i.e., when the height is equal to the diameter.