Linear Programming-Bellwork

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Presentation transcript:

Linear Programming-Bellwork Multiple Choice: What values of x and y minimize P for the objective function P = 3x + 2y? A. (2,0) B. ( 2,6) C. (4,3) D. (4,4)

Find the values of x and y that maximize P if P = –5x + 4y. Linear Programming Lesson 3-4 Additional Examples y – x + y x + y 3x – 11 2 3 1 4 11 > < Find the values of x and y that maximize P if P = –5x + 4y. Step 1: Graph the constraints. Step 2: Find the coordinates for each vertex. The solution is (1, 3), so A is at (1, 3). The solution is (5, 4), so B is at (5, 4). The solution is (4, 1), so C is at (4, 1).

Step 3: Evaluate P at each vertex. Linear Programming Lesson 3-4 Additional Examples (continued) Step 3: Evaluate P at each vertex. Vertex P = –5x + 4y A(1, 3) P = –5(1) + 4(3) = 7 B(5, 4) P = –5(5) + 4(4) = –9 C(4, 1) P = –5(4) + 4(1) = –16 When x = 1 and y = 3, P has its maximum value of 7. (1, 3)

Define: Let x = number of tables made in a day. Linear Programming Lesson 3-4 Additional Examples A furniture manufacturer can make from 30 to 60 tables a day and from 40 to 100 chairs a day. It can make at 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 per day to maximize profit? How much is the maximum profit? Tables Chairs Total No. of Products x y x + y No. of Units 30 x 60 40 y 100 120 Profit 150x 65y 150x + 65y Define: Let x = number of tables made in a day. Let y = number of chairs made in a day. Let P = total profit. Relate: Organize the information in a table. constraint objective <

Write: Write the constraints. Write the objective function. > < Linear Programming Lesson 3-4 Additional Examples (continued) x 30 x 60 y 40 y 100 x + y 120 > < Write: Write the constraints. Write the objective function. > < P = 150x + 65y < Step 1: Graph the constraints. Step 2: Find the coordinates of each vertex. Vertex A(30, 90) B(60, 60) C(60, 40) D(30, 40) Step 3: Evaluate P at each vertex. P = 150x + 65y P = 150(30) + 65(90) = 10,350 P = 150(60) + 65(60) = 12,900 P = 150(60) + 65(40) = 11,600 P = 150(30) + 65(40) = 7100 The furniture manufacturer can maximize their profit by making 60 tables and 60 chairs. The maximum profit is $12,900.

Find the values of x and y that maximize P if P = 3x + 2y. x 2 x 6 y 1 Linear Programming Lesson 3-4 Find the values of x and y that maximize P if P = 3x + 2y. x 2 x 6 y 1 y 5 x + y 8 > < > < <

Linear Programming Homework pg 142 1, 5, 7, 10