A rectangular shipping crate is to be constructed with a square base. The materi

Question

A rectangular shipping crate is to be constructed with a square base. The material for the two square ends costs $3 per square foot and the material for the sides costs $2 per square foot. What dimensions will minimize the cost of constructing the crate if it must have a volume of 12 cubic feet? What is the minimum cost? Let x be the length of the side of a square end, and y be the height of the crate. Be sure to check your answer. For what values of x and y will the right triangle inscribed in a semicircle of radius 2 have maximum area? What is the maximum area?

Explanation / Answer

(18) Total area of the square faces = 2x^2

Cost = 3(2x^2) = 6x^2

Total area of the side faces = 4xy

Cost = 2(4xy) = 8xy

Total cost = 6x^2 + 8xy

Volume = (x^2)y = 12

y = 12/(x^2)

Total cost C = 6x^2 + 8x(12/x^2) = 6x^2 + (96/x)

For minimum cost, dC/dx = 0

12x - (96/x^2) = 0

This gives x^3 = 8, so x = 2 ft and y = 12/x^2 = 3 ft

Minimum cost =  6(2)^2 + (96/2) = $72.

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