Question

Your iron works has been contracted to design and build a 500-ft3 square-based, open-top, rectangular steel holding tank for a paper company. The tank is to be made by welding half-inch-thick stainless steel plates together along their edges. As the production engineer, your job is to find dimensions for the base and height that will make the tank weight as little as possible. What dimensions do you tell the shop to use?

Answer 1

clearly the weight of the box is a function of the material used, which in turn depends on the surface area of the sheet metal used.
So we want a minimum surface area.
let each side of the base be x ft, and let the height be y ft.
Area = x^2 + 4xy
but x^2 y = 500
y = 500/x^2

area = x^2 + 4x(500/x^2) = x^2 + 2000/x
d(area)/dx = 2x -2000/x^2 = 0 for a min of area
2x = 2000/x^2
x^3 = 1000
x = 10 , then y = 500/10^2 = 5

the box should have a base of 10 ft by 10 ft, and a height of 5 ft.