A closed cylindrical can is to have a Surface Area S cubic units. Show that the can of maximum volume is achieved when the height is equal to the diameter of the base.

Question

A closed cylindrical can is to have a Surface Area S cubic units. Show that the can of maximum volume is achieved when the height is equal to the diameter of the base.

Answer 1

diameter, d = 2r

S = 2πr² + 2πrh V = πr²h

Solving for h from S: h = (S-2πr²)/(2πr) (Equation 1)

Substituting that into the volume formula:

V = πr²(S-2πr²)/(2πr) = Sr/2 - πr³

Taking the derivative of V and then setting to zero:

V ' = S/2 -3πr² = 0 --> S=6πr²

Substituting S into Equation 1 above:

h = (6πr² - 2πr²)/(2πr) = 4πr²/(2πr) = 2r = d, the diameter

Which finishes the proof.

Answer 2

Here's the formula: surface=r^2π*2+hdπ