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3-5

3-5. Linear Programming. Holt Algebra 2. Objective. Solve linear programming problems. Vocabulary. linear programming constraint feasible region objective function.

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3-5

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  1. 3-5 Linear Programming Holt Algebra 2

  2. Objective Solve linear programming problems.

  3. Vocabulary linear programming constraint feasible region objective function

  4. Linear programming is method of finding a maximum or minimum value of a function that satisfies a given set of conditions called constraints.A constraint is one of the inequalities in a linear programming problem. The solution to the set of constraints can be graphed as a feasible region.

  5. Example 1: Graphing a Feasible Region 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.

  6. Example 1 Continued Let x = the number of bread A, and y = the number of bread B. Write the constraints: x ≥ 0 The number of batches cannot be negative. y ≥ 0 The combined amount of oats is less than or equal to 180 pounds. 5x + 2y ≤ 180 The combined amount of flour is less than or equal to 135 pounds. 3x + 3y ≤ 135

  7. Graph the feasible region. 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. Check It Out! Example 1 Graph the feasible region for the following constraints. x ≥ 0 The number cannot be negative. y ≥ 1.5 The number is greater or equal to 1.5. 2.5x + 5y ≤ 20 The combined area is less than or equal to 20. 3x + 2y ≤ 12 The combined area is less than or equal to 12.

  9. Check It Out! Example 1 Continued Graph the feasible region. The feasible region is a quadrilateral with vertices at (0, 1.5), (0, 4), (2, 3), and (3, 1.5). Check A point in the feasible region, such as (2, 2), satisfies all of the constraints. 

  10. In most linear programming problems, you want to do more than identify the feasible region. Often you want to find the best combination of values in order to minimize or maximize a certain function. This function is the objective function. The objective function may have a minimum, a maximum, neither, or both depending on the feasible region.

  11. More advanced mathematics can prove that the maximum or minimum value of the objective function will always occur at a vertex of the feasible region.

  12. Example 2: Solving Linear Programming Problems 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.

  13. x ≥ 0 y ≥ 0 5x + 2y ≤ 180 3x + 3y ≤ 135 Example 2 Continued Step 1 Let P = the profit from the bread. Write the objective function: P = 40x + 30y Step 2 Recall the constraints and the graph from Example 1.

  14. Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. The maximum value occurs at the vertex (30, 15). Yum’s Bakery should make 30 batches of bread A and 15 batches of bread B to maximize the amount of profit.

  15. Helpful Hint Check your graph of the feasible region by using your calculator. Be sure to change the variables to x and y.

  16. x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 Check It Out! Example 2 Maximize the objective function P = 25x + 30y under the following constraints.

  17. x ≥ 0 y ≥ 1.5 2.5x + 5y ≤ 20 3x + 2y ≤ 12 Check It Out! Example 2 Continued Step 1 Write the objective function: P= 25x + 30y Step 2 Use the constraints to graph.

  18. Check It Out! Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. The maximum value occurs at the vertex (2, 3). P = 140

  19. Example 3: Problem-Solving Application Sue manages a soccer club and must decide how many members to send to soccer camp. It costs $75 for each advanced player and $50 for each intermediate player. Sue can spend no more than $13,250. Sue must send at least 60 more advanced than intermediate players and a minimum of 80 advanced players. Find the number of each type of player Sue can send to camp to maximize the number of players at camp.

  20. 1 Understand the Problem Example 3 Continued The answer will be in two parts—the number of advanced players and the number of intermediate players that will be sent to camp.

  21. 1 Understand the Problem • List the important information: • Advanced players cost $75. Intermediate players cost $50. • Sue can spend no more than $13,250. • Sue must send at least 60 more advanced players than intermediate players. • There needs to be a minimum of 80 advanced players. • Sue wants to send the maximum number of players possible.

  22. Make a Plan 2 Let x = the number of advanced players and y = the number of intermediate players. Write the constraints and objective function based on the important information. x ≥ 80 The number of advanced players is at least 80. The number of intermediate players cannot be negative. y ≥ 0 x – y ≥ 60 There are at least 60 more advanced players than intermediate players. The total cost must be no more than $13,250. 75x + 50y ≤ 13,250 Let P = the number of players sent to camp. The objective function is P = x + y.

  23. 3 Solve Graph the feasible region, and identify the vertices. Evaluate the objective function at each vertex. P(80, 0) = (80) + (0) = 80 P(80, 20) = (80) + (20) = 100 P(176, 0) = (176) + (0) = 176 P(130,70) = (130) + (70) = 200

  24. 4 Look Back Check the values (130, 70) in the constraints. x ≥ 80 y ≥ 0   130 ≥ 80 70 ≥ 0 x – y ≥ 60 75x + 50y ≤ 13,250 (130) – (70) ≥ 60 75(130) + 50(70) ≤ 13,250  60 ≥ 60 13,250 ≤ 13,250 

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