Abstract
The department secretary hands me a telephoned message from Rocky's Sand and Gravel. need a rectangular-based box that will hold of sand when filled level. The width must be half the length, and the box is to be open at the top. Material for the base of the box costs $4 per square foot while the material for the sides and ends costs $3 per square foot. Can you give me the dimensions of the minimum-cost box? Thanks. (signed) Rocky. It's the day before Easter recess, so hardly anyone has come to my calculus class. The four attending students and I agree to make a field trip to Rocky's Sand and Gravel. Rocky invites us inside his tiny office, graciously proffering coffee and cookies (sand tarts, naturally). We begin to show him how to formulate his problem using L, W, and H for the length, width, and height in feet. We draw a box and observe that the area of the base will be LW square feet. There are four other pieces: two ends each with area HW square feet and the two longer sides with areas HL square feet. Incorporating costs, we get the objective function: Cost in dollars = C = 4LW + 3(2 HW + 2 HL) =4LW+6H(L+ W). After Rocky informs us that three yards means 81 cubic feet, we write down the constraints: LWH=81 and W=0.5L. Substitution gives the cost as a function of length alone:
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