Haymarket Inc. is designing cardboard boxes for U-Haul. Market research has dete

Question

Haymarket Inc. is designing cardboard boxes for U-Haul.   Market research has determined that the following equation describes the amount customers will pay for boxes: 

$3 x Volume 0.5  where volume is measured in cubic feet

Haymarket’s cost for manufacturing boxes is $0.30 per square foot of surface area for oblong boxes (eight square corners, like a cartoon box), and $2.00 per square foot of surface area for other shapes. 

What should the shape and dimensions be of the box that Haymarket makes for U-Haul?  Use any optimization method you like.

Explanation / Answer

Since Haymarket’s cost for manufacturing boxes is $0.30 per square foot of surface area for oblong boxes (eight square corners, like a cartoon box), and $2.00 per square foot of surface area for other shapes and the selling price does not take shape into consideration, therefore, making oblong boxes will result in higher profit. Let the side of the oblong boxes made by Haymarket Inc be x ft. Then the surface are of the box is 6x2 so that its cost is 0.3*6x2 = 1.8x2. The volume of such a box is x3 so that its selling price is 3 (x3)0.5 = 3x1.5. Then the profit on sale of a box is P(x) = 3x1.5 -1.8x2. For maximum profit, we should have dP/dx = 0 and d2 P/dx2 should be negative. Here, dP/dx = 3*1.5x0.5 -1.8*2x = 4.5x0.5 -3.6x. Thus, if dP/dx = 0, then 4.5x0.5 = 3.6x or, 3.6x0.5 = 4.5 or x0.5 = 4.5/3.6 = 1.25 so that x = (1.25)2 or, x = 1.5625 ft. For this value of x,        d2 P/dx2 = 4.5*0.5x-0.5 – 3.6 = 2.25/1.25- 3.6 = 1.8-3.6 = -1.8 which is negative. Hence for optimizing profit, Haymarket Inc. should make oblong boxes of side 1.5625 ft for U-Haul .

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