1. Math 1 Warm Up In the Practice Workbook… Practice 7-6 (p. 94)
#2, 6, 7, 8, 10, 21

2. Math 1 Warm Up Answers

3. Math 1 Warm Up Answers

4. Math 1 Warm Up Answers

5. Math 1 Warm Up Answers
21a. 10x + 20y ≥ 40, 10x + 20y ≤ 60 x ≥ 0, y ≥ 0 21b.

6. Questions?

7. Linear Programming
Objective: To learn to find minimum and maximum values of a quantity.
linear programming – is a technique that identifies the minimum or maximum values of some quantity express as an equation of two or more variables.

8. Linear Programming
objective equation – is an equation of the quantity that you want to make as large or as small as possible.
constraints – are limits on the variables in the objective function written as linear inequalities.
feasible region – is the region of the coordinate plane that contains all the points that satisfy all the constraints.

9. Vertex Principle of Linear Programming
“If there is a maximum or a minimum value of the linear objective equation, it occurs at one or more of the vertices of the feasible region.”

10. Find the values of x and y that maximizes the objective function for the graph below. What is the maximum value? 1.

11. Find the values of x and y that minimizes the objective function for the graph below. What is the minimum value? 2.

12. Linear Programming Steps: Graph the constraints.
Find the coordinates for each vertex of the feasible region.
Evaluate the objective equation at each vertex.

13. Constraints: x ≤ 8 y ≤ 5 x + y ≤ 2 Objective: C = x + 3y
Find the values of x and y that minimizes the objective equation C. What is the minimum value of C?
Constraints: x ≤ 8 y ≤ 5 x + y ≤ 2 Objective: C = x + 3y

15. Constraints: x ≥ 0 y ≥ 0 x ≤ 5 y ≤ 4 Objective: P = 3x + 2y
Find the values of x and y that maximizes the objective equation P. What is the maximum value of P?
Constraints: x ≥ 0 y ≥ 0 x ≤ 5 y ≤ 4 Objective: P = 3x + 2y

16. Constraints: x ≥ 0 y ≥ 5 x + y ≥ 8 Objective: F = 2x + 5y
Find the values of x and y that minimizes the objective equation F. What is the minimum value of F?
Constraints: x ≥ 0 y ≥ 5 x + y ≥ 8 Objective: F = 2x + 5y

17. Find the values of x and y that maximizes the objective equation P
Find the values of x and y that maximizes the objective equation P. What is the maximum value of P?
Constraints: x ≥ 0 y ≥ 0 y ≥ 3 2 x – 3 y ≤ -x + 7 Objective: P = 4x + y

18. Find the values of x and y that maximizes the objective equation N
Find the values of x and y that maximizes the objective equation N. What is the maximum value of N?
Constraints: x ≥ 0 y ≥ 0 2x + y ≥ 10 x + y ≤ 8 Objective: N = 100x + 40y

19. Apply!
A furniture manufacture can make from 30 to 60 tables per day and 40 to 100 chairs per day. The manufacture can make a most 120 units in one day. The profit on a table is $150, and the profit on a chair is $65. How many tables and chairs should they make to maximize profit? How much is the maximum profit?

20. Apply!
Jerry works no more than 20 hours a week during the school year. He is paid $10 an hour for tutoring math students and $17 an hour for delivering pizzas for Pizza King. He wants to spend at least 3 hours but no more than 8 hours a week tutoring. How many hours should Jerry work at each job to maximize his weekly earning? Find Jerry’s maximum weekly earning.

21. Assignment
Linear Programming
Handout