math
posted by ibranian .
Your iron works has been contracted to design and build a 500ft3 squarebased, opentop, rectangular steel holding tank for a paper company. The tank is to be made by welding halfinchthick 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?

math 
Reiny
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.
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