A manufacturing company wants to package its product in a rectangular box with a

Question

A manufacturing company wants to package its product in a rectangular box with a square base and a volume of 32 cubic inches. The cost of the material used for the top is $.01 square inch; the cost of the material used for the bottom is $.03 per square inch, and the cost of the material used for four sides is $.02 per square inch. What are the dimensions of the box with the minimum cost? Show work

Explanation / Answer

Let y be the height of the box and x be the length/width. Then the cost c is: C = .01x^2 + .03x^2 + 4(.02)xy = .04x^2 + .08xy But V = (x^2)y = 32 => y = 32/x^2 So we have C = .04x^2 + .08x(32/x^2) = .04x^2 + 2.56x^-1 C' = .08x - 2.56x^-2 0 = .08x - 2.56x^-2 .08x = 2.56x^-2 .08x^3 = 2.56 x^3 = 32 x = 3.17 => y(3.17)^2 = 32 y = 32/(3.17^2) = 3.18

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