A manufacturer wants to procure rectangular boxes to ship its product. The boxes must contain 20 cubic feet of space. To be durable enough to ensure the safety of the product, the material for the sides of the boxes will cost $0.10 per square foot, while the material for the top and bottom will cost $0.25 per square foot. In this activity we will help the manufacturer determine the box of minimal cost.
What quantities are constant in this problem? What are the variables in this problem? Provide appropriate variable labels. What, if any, restrictions are there on the variables?
Using your variables from (a), determine a formula for the total cost of a box.
Your formula in part (b) might be in terms of three variables. If so, find a relationship between the variables, and then use this relationship to write as a function of only two independent variables.
Find the dimensions that minimize the cost of a box. Be sure to verify that you have a minimum cost.
Long, Keller & Schlicker, Active Calculus Multivariable, §11.10 Optimization, Exercise 37