3-4 Linear Programming

Adjusting the Window To adjust the viewing window, press window Xmin = smallest x-value Xmax = largest x-value Xscl = # of units represented by tick marks on the x-axis Ymin = smallest y-value Ymax = largest y-value Yscl = # of units represented by tick marks on the y-axis

Linear Programming y – x + > 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 and minimize P if P = –5x + 4y. Step 1: Graph the constraints. Step 2: Find the coordinates for each vertex. To find A, solve the system . The solution is (1, 3), so A is at (1, 3). y = – x + y = x + 2 3 11 1 4 To find B, solve the system . The solution is (5, 4), so B is at (5, 4). y = x + y = 3x – 11 1 4 11

Linear Programming y = – x + y = 3x – 11 To find C, solve the system . Lesson 3-4 Additional Examples (continued) To find C, solve the system . The solution is (4, 1), so C is at (4, 1). y = – x + y = 3x – 11 2 3 11 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. When x = 4 and y = 1, P has its minimum value of –16.

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 <

Linear Programming x 30 x 60 y 40 y 100 x + y 120 > < 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.