#### Some Optimization Problems

• An artist is planning to sell signed prints of her latest work. If 50 prints are offered for sale, she can charge \$400 each. However, if she makes more than 50 prints, she must lower the price of all the prints by \$5 for each print in excess of the 50. How many prints should the artist make in order to maximize her revenue?

(Source: Calculus and Its Applications, by Goldstein, Lay and Schneider, Section 2.7 Exercise 13.)

• A rectangular corral of 54 square meters is to be fenced off and then divided by a fence into two sections. Find the dimensions of the corral so that the amount of fencing required is minimized.

(Source: Calculus and Its Applications, by Goldstein, Lay and Schneider, Section 2.6 Exercise 13.)

• What dimensions (radius and height, in centimeters) should be given to a cylindrical can so that it holds exactly 1 liter (= 1000 cm3) and so that the amount of aluminum contained in the can is minimized?

(Source: Thomas' Calculus, by Finney, Weir and Giordano, Section 3.5 Example 2.)

• As above, we seek a cylindrical can holding 1 liter. It has come to our attention that, in the manufacturing process, the circles for the top and bottom of the can are cut from squares, with the excess being thrown out as scrap. If we wish to minimize the amount of aluminum used (accounting for the scrap as well), what dimensions should be used?

(Source: Thomas' Calculus, by Finney, Weir and Giordano, Section 3.5 Exercise 15.)

• A furniture store expects to sell 640 sofas at a steady rate next year. The manager of the store plans to order these sofas from the manufacturer by placing several orders of the same size spaced equally throughout the year. The ordering cost for each delivery is \$160, and carrying costs, based on the average number of sofas in inventory, amount to \$32 per year for one sofa. Find the inventory cost in terms of the order quantity x and the number r of orders placed during the year. Then determine the economic order quantity.

(Source: Calculus and Its Applications, by Goldstein, Lay and Schneider, Section 2.6 Exercise 4.)

• Foggy Optics, Inc. makes laboratory microscopes. Setting up each production run costs \$2500. Insurance costs, based on the average number of microscopes in the warehouse, amount to \$20 per microscope per year. Storage costs, based on the maximum number of microscopes in the warehouse, amount to \$15 per microscope per year. Suppose that the company expects to sell 1600 microscopes at a fairly uniform rate throughout the year. Determine the number of production runs that will minimize the company's overall expenses.

(Source: Calculus and Its Applications, by Goldstein, Lay and Schneider, Section 2.6 Exercise 7.)

• The average ticket price for a concert at the opera house was \$50. The average attendance was 4000. When the ticket price was raised to \$52, attendance declined to an average of 3800 persons per performance. What should the ticket price be in order to maximize the revenue for the opera house? (Assume a linear demand curve.)

(Source: Calculus and Its Applications, by Goldstein, Lay and Schneider, Section 2.7 Exercise 12.)

This page maintained by: Thomas L. Scofield
Department of Mathematics and Statistics, Calvin College