SECTION 11.8 Linear Programming 845 Concepts and Vocabulary 11.8 Assess Your Understanding 1. A linear programming problem requires that a linear expression, called the , be maximized or minimized. 2. True or False If a linear programming problem has a solution, it is located at a corner point of the graph of the feasible points. Skill Building In Problems 3–8, find the maximum and minimum value of the given objective function of a linear programming problem. The figure illustrates the graph of the feasible points. 3. = + z x y 4. = + z x y 2 3 5. = + z x y 10 6. = + z x y 10 7. = + z x y 5 7 8. = + z x y 7 5 In Problems 9–18, solve each linear programming problem. 9. Maximize = + z x y 2 subject to the constraints ≥ ≥ + ≤ + ≥ x y x y x y 0, 0, 6, 1 10. Maximize = + z x y3 subject to the constraints ≥ ≥ + ≥ ≤ ≤ x y x y x y 0, 0, 3, 5, 7 11. Minimize = + z x y 2 5 subject to the constraints ≥ ≥ + ≥ ≤ ≤ x y x y x y 0, 0, 2, 5, 3 12. Minimize = + z x y 3 4 subject to the constraints ≥ ≥ + ≥ + ≤ x y x y x y 0, 0, 2 3 6, 8 13. Maximize = + z x y 3 5 subject to the constraints ≥ ≥ + ≥ + ≤ + ≤ x y x y x y x y 0, 0, 2,2 3 12,3 2 12 14. Maximize = + z x y 5 3 subject to the constraints ≥ ≥ + ≥ + ≤ + ≤ x y x y x y x y 0, 0, 2, 8, 2 10 15. Minimize = + z x y 5 4 subject to the constraints ≥ ≥ + ≥ + ≤ + ≤ x y x y x y x y 0, 0, 2, 2 3 12, 3 12 16. Minimize = + z x y 2 3 subject to the constraints ≥ ≥ + ≥ + ≤ + ≥ x y x y x y x y 0, 0, 3, 9, 3 6 17. Maximize = + z x y 5 2 subject to the constraints ≥ ≥ + ≤ + ≥ + ≥ x y x y x y x y 0, 0, 10, 2 10, 2 10 18. Maximize = + z x y 2 4 subject to the constraints ≥ ≥ + ≥ + ≤ x y x y x y 0, 0, 2 4, 9 x y 8 8 5 (5, 6) (4, 0) (5, 2) (0, 3) 24 21 (0, 6) 1. Now Work 1. Modeling 1.ExplainingConcepts Calculus Preview 1.InteractiveFigure 19. Maximizing Profit A manufacturer of skis produces two types: downhill and cross-country. Use the following table to determine how many of each kind of ski should be produced to achieve a maximum profit. What is the maximum profit? What would the maximum profit be if the time available for manufacturing were increased to 48 hours? Downhill Crosscountry Time Available Manufacturing time per ski 2 hours 1 hour 40 hours Finishing time per ski 1 hour 1 hour 32 hours Profit per ski $70 $50 20. Farm Management A farmer has 70 acres of land available for planting either soybeans or wheat. The cost of preparing the soil, the workdays required, and the expected profit per acre planted for each type of crop are given in the following table. Soybeans Wheat Preparation cost per acre $60 $30 Workdays required per acre 3 4 Profit per acre $180 $100 Applications and Extensions (continued)
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