A rectangular box with a square base and cover is to have a volume of 2500 cubic feet. If the cost per square foot for the bottom is $2, for the top is $3, and for the sides is $1, what should the dimensions be in order to minimize the cost?

Answer 1

If the base has side x, and the height is h, then

x^2h = 2500
h = 2500/x^2

So, the cost to make the box is

c = 2(x^2) + 3(x^2) + 1*4*xh
= 5x^2 + 10000/x
dc/dx = 10x-10000/x^2
dc/dx=0 when x=10

so, the box is 10x10x25 ft