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

Question

Philoid · Accepted Answer

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

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

S = 2πr² + 2πrh

⇒ h

Let V be the volume of the cylinder. Then

V = πr²h

Now,

If

So, when then <0

Then, by second derivative test, the volume is the maximum when

Now, when . then h =

Therefore, the volume is the maximum when the height is the twice the radius or height is equal to diameter.