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3.4 Pp

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3.4 Pp

1. 1. <ul><li>Bell Ringer: </li></ul><ul><li>Graph the system of inequalities from problem # 36 on your homework last night. </li></ul><ul><li>2. Is (-1,1) a solution of the system? </li></ul><ul><li>Why or why not? </li></ul>
2. 2. Solve linear programming problems. Objective
3. 3. optimization linear programming constraint feasible region New Vocabulary
4. 4. Check It Out! Example 1 Graph the feasible region for the following constraints. (Hint: Find the vertices) x ≥ 0 y ≥ 1.5 2.5 x + 5 y ≤ 20 3 x + 2 y ≤ 12
5. 5. Yum’s Bakery bakes two breads, A and B . One batch of A uses 5 pounds of oats and 3 pounds of flour. One batch of B uses 2 pounds of oats and 3 pounds of flour. The company has 180 pounds of oats and 135 pounds of flour available. Write the constraints for the problem and graph the feasible region. Example 1: Graphing a Feasible Region
6. 6. Graph the feasible region.
7. 7. The feasible region is a quadrilateral with vertices at (0, 0), (36, 0), (30, 15), and (0, 45). Check A point in the feasible region, such as (10, 10), satisfies all of the constraints. 
8. 8. Why would we want to find a feasible region? objective function:
9. 11. Yum’s Bakery wants to maximize its profits from bread sales. One batch of A yields a profit of \$40. One batch of B yields a profit of \$30. Use the profit information and the data from Example 1 to find how many batches of each bread the bakery should bake. Example 2: Solving Linear Programming Problems
10. 12. Example 2 Continued Step 1 Let P = the profit from the bread. Write the objective function: P = 40 x + 30 y Step 2 Recall the constraints and the graph from Example 1. x ≥ 0 y ≥ 0 5 x + 2 y ≤ 180 3 x + 3 y ≤ 135
11. 13. Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. Yum’s Bakery should make 30 batches of bread A and 15 batches of bread B to maximize the amount of profit. ( x , y ) 40 x + 30 y P(\$) (0, 0) 40(0) + 30(0) (0, 45) (30, 15) (36, 0)
12. 14. Ticket out the Door Use your own words to define Vertex Principle of Linear Programming.