Expert: Abe Mantell - 12/28/2006

Question
A box with a square base is needed to package 100cm^3 of powdered milk.  Find the dimensions of the box that will use the minimum amount of material.

Answer
Let x = the height of the box
and y = the length of one edge of the square base

Thus, y*x^2=100 ==> y=100/x^2
Minimize the total area, A = x^2 + 4*x*y
A(x) = x^2 + 4x(100/x^2) = x^2 + 400/x
Now solve, A'(x)=0 ==> 2x-400/x^2 = 0 ==> x=2(5)^(2/3) cm
or x is about 5.848 cm, and y is about 2.924 cm

Abe