Calculus

A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs \$2 per square foot and the metal for the sides costs \$7 per square foot. Find the dimensions that minimize cost if the box has a volume of 30 cubic feet.

Length of base = ?
Height of side z = ?

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1. side of top and bottom = x
height = y

area of top and bottom together= 2 x^2
cost of top and bottom together = 4 x^2

area of sides = 4 x y
cost of sides = 28 x y

Total cost c = 4 x^2 + 28 x y

Volume = x^2 y = 30
so y = 30/x^2

c = 4 x^2 + 28 x (30/x^2)
c = 4 x^2 + 840/x
dc/dx = 0 for min or max
0 = 8 x - 840/x^2
x = 105/x^2
x^3 = 105
x = 4.72
then
y = 30/x^2 = 1.35

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