a cylindrical soup can has a volume of 335cm^3.find the dimensions(radius r and height h)that minimise the surface area of such a can.

Answer 1

V = πr^2 h
h = 335/(π r^2)

SA = 2π r^2 + 2πrh
= 2πr^2 + 2πr(335/(πr^2)
= 2πr^2 + 670/r

d(SA)/dr = 4πr - 670/r^2
= 0 for a min of SA

4πr = 670/r^2
4πr^3 = 670
r = (670/(4π))^(1/3) = 3.763757606..
subbing back into h
h = 7.5275152..

notice that h : r = 2 : 1
that is, the height should be equal to the diameter for a minimum surface area