posted by shawn .
A rectangular box with a square base and top is to be made to contain 1250 cubic feet. The material for the base costs 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.
Your top and bottom, averaged, are more expensive than the sides. Top and bottom: 15+35=50/2 = 25cents per square foot. Two sides: 20+20=40/2=20cents per square foot. therefore, you would want as little area on the bases as possible. If you place one square foot each on the top and bottom, that leaves 1248 square feet for the sides. 1248/4 = 312 square feet per side. The sides are one foot (length of base) x 312. Therefore, your box would be 312x1x1. This would be (1 sq ft x 15 cents) + (1 sq ft x 35 cents) + 1248 sq ft x 20 cents)
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