A box with a square base and open top must have a volume of 62500 cm cubed. We wish to find the dimensions of the box that minimize the amount of material used.
First, find a formula for the surface area of the box in terms of only x, the length of one side of the square base.
Next, find the derivative.
my SA = x^2 + 4xy
given : x^2y =62500 ----> y = 62500/x^2
SA = x^2 + 4x(62500/x^2 = x^2 + 250,000/x
d(SA)/dx = 2x - 250,000/x^2
= 0 for a min of SA
x^3 = 125,000
x = 50 , then y = 62500/2500 = 25