College Algebra 9th txtbk

28 CHAPTER 7 SYSTEMS OF EQUATIONS AND MATRICES
8. Can a linear programming problem have more than one optimal
value? Explain.
In Problems 9–12, find the maximum value of each objective
function over the feasible region S shown in the figure.
9. z x y 10. z 4x y
11. z 3x 7y 12. z 9x 3y
z max 90 at (7, 9) and (10, 0) (multiple
optimal solutions)
Problems 13–16 refer to the feasible region S shown and the
constant-profit lines discussed in Explore-Discuss 1. For each
objective function, draw the line that passes through the feasible
point (5, 5) and use the straightedge method from Explore-Discuss
1 to find the maximum value. Check your answer by evaluating the
objective function at each corner point.
Maximum value of z on S is
13. z x 2y 14. z 3x y
30 at both (7, 9) and (10, 0)
15. z 7x 2y 16. z 2x 8y Maximum value of z on S is
Maximum value of z
96 at (0, 12)
on S is 70 at (10, 0).
In Problems 17–20, find the minimum value of each objective
function over the feasible region T shown in the figure.
(0, 8)
5
y
5
(0, 0)
(0, 12)
(4, 3)
17. z 7x 4y 18. z 7x 9y
y
(0, 12)
T
5
S
5
(7, 9)
(10, 0)
x
z max 40 at (10, 0)
(12, 0)
19. z 3x 8y 20. z 5x 4y
z min 32 at (0, 8) and (4, 3)
(multiple optimal solutions)
Problems 21–24 refer to the feasible region T shown for problems
17–20. For each objective function, draw the constant-value line
that passes through the feasible point (5, 5) and use the
straightedge method from Explore-Discuss 1 to find the minimum
x
z min 55 at (4, 3)
value. Check your answer by evaluating the objective function at
each corner point.
Minimum value of z on
21. z x 2y 22. z 2x y
T is 8 at (0, 8).
23. z 5x 4y 24. z 2x 8y
Minimum value of z on T is Minimum value of z on
32 at both (0, 8) and (4, 3). T is 24 at (12, 0).
B In Problems 25–38, solve the linear programming problems.
25. Maximize
Subject to
26. Maximize
Subject to
27. Minimize
Subject to
28. Minimize
Subject to
29. Maximize
Subject to
30. Maximize
Subject to
31. Minimize
Subject to
z 3x 2y
x 2y 10
3x y 15
x, y 0
z 4x 5y
2x y 12
x 3y 21
x, y 0
z 3x 4y
2x y 8
x 2y 10
x, y 0
z 2x y
4x 3y 24
4x y 16
x, y 0
z 3x 4y
x 2y 24
x y 14
2x y 24
x, y 0
z 5x 3y
3x y 24
x y 10
x 3y 24
x, y 0
z 5x 6y
x 4y 20
4x y 20
x y 20
x, y 0
Maximum value of z on
S is 18 at (4, 3).
z max 42 at (3, 6)
Minimum value of z on
S is 12 at (4, 0).
z min 8 at (0, 8)
Maximum value of z on
S is 52 at (4, 10).
z max 44 at (7, 3)
Minimum value of z on
S is 44 at (4, 4).
32. Minimize z x 2y
Subject to 2x 3y 30
3x 2y 30
x y 15
x, y 0 z min 15 at (15, 0)
33. Minimize and maximize z 25x 50y
Subject to x 2y 120
The minimum value of z on S is
x y 60
1,500 at (60, 0). The maximum value
x 2y 0 of z on S is 3,000 at (60, 30) and
x, y 0 (120, 0) (multiple optimal solutions).
34. Minimize and maximize z 15x 30y
Subject to x 2y 100
2x y 0
z max 6000 at (0, 200);
2x y 200 z min 1500 at (0, 50) and (20, 40)
x, y 0 (multiple optimal solutions)