3-4 Linear Programming Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

Determine if the given ordered pair is a solution of Warm Up Determine if the given ordered pair is a solution of x + y ≥ 6 x – 2y >10 2. (10, 1) no 1. (3, 3) no 4. (15, 2) yes 3. (12, 0) yes

Objective Solve linear programming problems.

Vocabulary linear programming constraint feasible region objective function

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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.

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.

Let x = the number of bread A, and y = the number of bread B. 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

Graph the feasible region 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. 

Graph the feasible region for the following constraints. 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.

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. 

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.

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.

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.

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. x ≥ 0 y ≥ 0 5x + 2y ≤ 180 3x + 3y ≤ 135

Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. (x, y) 40x + 30y P($) (0, 0) 40(0) + 30(0) (0, 45) 40(0) + 30(45) 1350 (30, 15) 40(30) + 30(15) 1650 (36, 0) 40(36) + 30(0) 1440 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.

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

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

Check It Out! Example 2 Continued Step 3 Evaluate the objective function at the vertices of the feasible region. (x, y) 25x + 30y P($) (0, 4) 25(0) + 30(4) 120 (0, 1.5) 25(0) + 30(1.5) 45 (2, 3) 25(2) + 30(3) 140 (3, 1.5) 25(3) + 30(1.5) The maximum value occurs at the vertex (2, 3). P = 140

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.

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

Understand the Problem 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.

2 Make a Plan 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.

Solve 3 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

Check the values (130, 70) in the constraints. Look Back 4 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 

Check It Out! Example 3 A book store manager is purchasing new bookcases. The store needs 320 feet of shelf space. Bookcase A provides 32 ft of shelf space and costs $200. Bookcase B provides 16 ft of shelf space and costs $125. Because of space restrictions, the store has room for at most 8 of bookcase A and 12 of bookcase B. How many of each type of bookcase should the manager purchase to minimize the cost?

Understand the Problem 1 Understand the Problem The answer will be in two parts—the number of bookcases that provide 32 ft of shelf space and the number of bookcases that provide 16 ft of shelf space. List the important information: Bookcase A cost $200. Bookcase B cost $125. The store needs at least 320 feet of shelf space. Manager has room for at most 8 of bookcase A and 12 of bookcase B. Minimize the cost of the types of bookcases.

2 Make a Plan Let x represent the number of Bookcase A and y represent the number of Bookcase B. Write the constraints and objective function based on the important information. x ≥ 0 The number of Bookcase A cannot be negative. y ≥ 0 The number of Bookcase B cannot be negative. x ≤ 8 There are 8 or less of Bookcase A. y ≤ 12 There are 12 or less of Bookcase B. 32x + 16y ≤ 320 The total shelf space is at least 320 feet. Let P = The number of Bookcase A and Bookcase B. The objective function is P = 200x + 125y.

Solve 3 Graph the feasible region, and identify the vertices. Evaluate the objective function at each vertex. P(4, 12) = (800) + (1500) = 2300 P(8, 12) = (1600) + (1500) = 3100 P(8, 4) = (1600) + (500) = 2100

Look Back 4 Check the values (8, 4) in the constraints. x ≥ 0 y ≥ 0 x ≤ 8 y ≤ 12 8 ≥ 0  4 ≥ 0  8 ≤ 8  4 ≤ 12  32x + 16y ≤ 320 32(8) + 16(4) ≤ 320 256 + 64 ≤ 320 320 ≤ 320 

Lesson Quiz 1. Ace Guitars produces acoustic and electric guitars. Each acoustic guitar yields a profit of $30, and requires 2 work hours in factory A and 4 work hours in factory B. Each electric guitar yields a profit of $50 and requires 4 work hours in factory A and 3 work hours in factory B. Each factory operates for at most 10 hours each day. Graph the feasible region. Then, find the number of each type of guitar that should be produced each day to maximize the company’s profits.

Lesson Quiz 1 acoustic; 2 electric