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Question

A supermarket is to be designed as a rectangular building with a floor area of 12,000 square feet. The front of the building will be mostly glass and will cost $70 per running foot for the materials.The other three walls will be constructed of cement blocks at a cost of $50 per running foot. Ignore all other costs and find the dimensions of the base of the building that will minimize the cost of the materials for the four walls of the building, rounding each to the nearest 0.1 foot. Clearly label the dimensions and justify that this gives the minimum cost by showing the appropriate calculus work.

Answer 1

Area l*w = 12,000, so l = 12,000/w. Cost, arbitrarily assuming w is front dimension, is C = 70w + 50w + 2*50*l = 120w + 1,200,000 w^-1. dC/dw = 120 - 1,200,000 w^-2 --> zero for an extremum, so w² = 1,200,000 / 120 = 10⁴ and w = 100.0 ft, l = 120.0 ft. Resulting cost is $24,000, and is confirmed to be a minimum by the second derivative 2,400,000 w^-3 being positive.