6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 375 times and does not want more than $12,000 in inventory at any one time. a) Using A to represent Apple tablets and S to represent Samsung tablets, translate the problem into a system of linear inequalities. A S A S A S 2 , 10, 5, 300 200 12,000 ≥ ≥ ≥ + ≤ b) Graph the system of linear inequalities and indicate the solution set. Graph the number of Apple tablets on the horizontal axis and the number of Samsung tablets on the vertical axis * c) Select a point in the solution set and determine the inventory cost for the two models that corresponds to the point. One example is 25 Apple tablets and 12 Samsung tablets at an inventory cost of $9900. In Exercises 51–56, solve the linear programming problems. 51. MODELING—On Wheels The Boards and Blades Company manufactures skateboards and in-line skates. The company can produce a maximum of 20 skateboards and pairs of in-line skates per day. It makes a profit of $50 on a skateboard and a profit of $40 on a pair of in-line skates. The company’s planners want to make at least 3 skateboards but not more than 6 skateboards per day. To keep customers happy, they must make at least 2 pairs of in-line skates per day. a) List the constraints. x y x x y 20, 3, 6, 2 + ≤ ≥ ≤ ≥ b) Determine the objective function. P x y 50 40 = + c) Graph the set of constraints. * d) Determine the vertices of the feasible region. (3, 17), (6, 14), (6, 2), (3, 2) e) How many skateboards and pairs of in-line skates should be made to maximize the profit? Six skateboards and 14 pairs of in-line skates f) Determine the maximum profit. $860 52. MODELING—Washing Machine Production A company manufactures two types of washers: top load and front load. The company can manufacture a maximum of 18 washers per day. It makes a profit of $125 on top load machines and $200 on front load machines. No more than 5 front load machines can be manufactured due to production restrictions. To meet consumer demand, the company must manufacture at least 2 front load machines and 2 top load machines per day. a) List the constraints. x y y x y 18, 5, 2, 2 + ≤ ≤ ≥ ≥ b) Determine the objective function. P x y 125 200 = + $ $ c) Graph the set of constraints. * d) Determine the vertices of the feasible region. (2, 5), (13, 5), (16, 2), (2, 2) e) How many washing machines of each type should be made to maximize profit? 13 top load machines and 5 front load machines f) Determine the maximum profit. $2625 53. MODELING—Paint Production A paint supplier has two machines that produce both indoor paint and outdoor paint. To meet one of its contractual obligations, the company must produce at least 60 gal of indoor paint and 100 gal of outdoor paint. Machine I makes 3 gal of indoor paint and 10 gal of outdoor paint per hour. Machine II makes 4 gal of indoor paint and 5 gal of outdoor paint per hour. It costs $28 per hour to run machine I and $33 per hour to run machine II. a) List the constraints. x y x y x y 3 4 60, 10 5 100, 0, 0 + ≥ + ≥ ≥ ≥ b) Determine the objective function. C x y 28 33 = + c) Graph the set of constraints. * d) Determine the vertices of the feasible region. (0, 20), (4, 12), (20, 0) e) How many hours should each machine be operated to fulfill the contract at a minimum cost? 4 hours for Machine I and 12 hours for Machine II f) Determine the minimum cost. $508 54. MODELING—Special Diet A dietitian prepares a special diet using two food groups, A and B. Each ounce of food group A contains 3 units of vitamin C and 1 unit of vitamin D. Each ounce of food group B contains 1 unit of vitamin C and 2 units of vitamin D. The minimum daily requirements with this diet are at least 9 units of vitamin C and at least 8 units of vitamin D. Each ounce of food group A costs 50 cents, and each ounce of food group B costs 30 cents. a) List the constraints. x y x y x y 3 9, 2 8, 0, 0 + ≥ + ≥ ≥ ≥ b) Determine the objective function for minimizing cost. C x y 0.50 0.30 = + c) Graph the set of constraints. * d) Determine the vertices of the feasible region. (0, 9), (2, 3), (8, 0) e) How many ounces of each food group should be used to meet the daily requirements and minimize the cost? 2 ounces of food group A and 3 ounces of food group B f) Determine the minimum cost. $1.90 Challenge Problems/Group Activities 55. MODELING—Hot Dog Profits To make one pack of all-beef hot dogs, a manufacturer uses 1 lb of beef; to make one pack of regular hot dogs, the manufacturer uses lb 1 2 each of beef and pork. The profit on the allbeef hot dogs is 40 cents per pack, and the profit on regular hot dogs is 30 cents per pack. If there are 200 lb $ $ *See Instructor Answer Appendix Lzf/Shutterstock
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