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

Let the radius of base circle and top circle of cylinder be r and h be the height of cylinder

V be the volume of cylinder and S be the surface area which is constant

We have to find h such that volume is maximum

Surface area of cylinder is given by

⇒ S = 2πr² + 2πrh

Volume of cylinder is given by

⇒ V = πr²h

⇒ V = 1/2(Sr – 2πr³)

We have to maximise V that is find value of h where V is maximum

For that first we need to find the critical points where

Differentiate V with respect to r and equate to 0

⇒ 0 = 1/2(S – 6πr²)

⇒ S = 6πr²

But S = 2πr² + 2πrh

⇒ 2πr² + 2πrh = 6πr²

⇒ 2πrh = 4πr²

⇒ h = 2r

Now we have to check whether h = 2r is a point of maxima or minima. For h = 2r to be a point of maxima we should have

Let us check

We have as

Differentiate again with respect to r

r is radius hence it is positive always hence -6πr is negative hence

Hence h = 2r is a point of maxima

2r is diameter

Hence proved maximum volume is attained when height is equal to diameter