278 CHAPTER 5 Linear Inequalities and Linear Programming CONCEPTUAL INSIGHT SECTION 5.3 Linear Programming in Two Dimensions: A Geometric Approach 279 Refer to Example 3. If we change the minimum requirement for drug B from 120 to 19. Minimize and maximize 125, the optimal solution changes to 3.6 grams of substance M and 24.1 grams of suh z = 2x + 3y 28. Minimize and maximize subject to 2x + y = 10 P = -x+ 3y stance N, correct to one decimal place. Now refer to Example 1. If we change the maximum labor-hours available per day x + 2y 2 8 subject to 2x - y = 4 in the assembly department from 84 to 79, the solution changes to 15 standard tents x, y 2 0 -x + 2y $ 4 and 8.5 expedition tents. 20. Minimize and maximize ys 6 We can measure 3.6 grams of substance M and 24.1 grams of substance N, but z = &x + Ty r, y 20 how can we make 8.5 tents? Should we make 8 tents? Or 9 tents? If the solutions to subject to 4x + 3y 2 24 29. Minimize and maximize a problem must be integers and the optimal solution found graphically involves deci- 3x + 4y 2 8 P = 20x + 10y mals, then rounding the decimal value to the nearest integer does not always produce .x, y Z subject to 2x + 3y 2 30 the optimal integer solution (see Problem 44, Exercises 5.3). Finding optimal integer 21. Maximize P = 30x + 40y 2r + y s 26 solutions to a linear programming problem is called integer programming and requires -2x + 5y $ 34 special techniques that are beyond the scope of this book. As mentioned earlier, if we subject to 2x + y s 10 encounter a solution like 8.5 tents per day, we will interpret this as an average value *+ ys 7 t, y 2 0 over many days of production. r + 2y s 12 30. Minimize and maximize x, y 2 0 P = 12x + 14y 22. Maximize P = 20x + 10y subject to -2x + y 2 6 subject to 3x + y s 21 xtys 15 Exercises 5.3 x + ys 9 3.x - y = 0 x + 3y $ 21 x, y 2 0 W Skills Warm-up Exercises 9. P = xty 10. P = 4x + y x, y ZO 31. Maximize P = 20.x + 30y subject to In problems 1-8, if necessary, review Theorem 1. In Problems 0.6.x + 1.2y s 960 1-4, the feasible region is the set of points on and inside the 11. P = 3x + 7y 12. P = 9x + 3y 23. Minimize and maximize z = 10x + 30y 0.03.x + 0.04y s 36 rectangle with vertices (0, 0). (12, 0). (0, 5), and (12, 5). In Problems 13-16, graph the constant-cost lines through (9, 9) subject to 2r + y 2 16 0.3.x + 0.2y = 270 Find the maximum and minimum values of the objective and (12, 12). Use a straightedge to identify the corner point where x + y z 12 x , y 2 function Q over the feasible region. the minimum cost occurs. Confirm your answer by constructing a corner-point table. x + 2y 2 14 32. Minimize C = 30x + 10y 1. Q = 7x + 14y 2. Q = 3x + 15y x, y z 0 subject to 1.8.x + 0.9y 2 270 3. Q = 10r - 12y 4. Q = -9x + 20y 0.3.x + 0.2y 2 54 (15, 15) 24. Minimize and maximize (0, 15) z = 400x + 100y 0.01.r + 0.03y 2 3.9 In Problems 5-8, the feasible region is the set of points on and inside the triangle with vertices (0. 0), (8, 0), and (0, 10). (12, 12) subject to 3x + y 2 24 Find the maximum and minimum values of the objective func- x + y z 16 33. Maximize P = 525x + 478y 10 - tion Q over the feasible region. 0.8) x + 3y 2 30 subject to 275x + 322y