Ch. 3.4 I can solve problems using linear programming Do Now: 1. Graph the system of inequalities. y ≤ x – 2 y ≥ –2x – 2 x ≤ 4 x ≥ 1 Today’s Agenda Success Criteria: Check assignment Do Now Lesson Assignment Test verticies Use programming to maximize profit

Linear Programming Businesses use linear programming to find out how to maximize profit or minimize costs. Most have constraints on what they can use or buy.

Example 2: Art Application Lauren wants to paint no more than 70 plates for the art show. It costs her at least $50 plus $2 per item to produce red plates and $3 per item to produce gold plates. She wants to spend no more than $215. Write and graph a system of inequalities that can be used to determine the number of each plate that Lauren can make.

Example 2 Continued Let x represent the number of red plates, and let y represent the number of gold plates. The total number of plates Lauren is willing to paint can be modeled by the inequality x + y ≤ 70. The amount of money that Lauren is willing to spend can be modeled by 50 + 2x + 3y ≤ 215. x  0 y  0 The system of inequalities is . x + y ≤ 70 50 + 2x + 3y ≤ 215

Example 2 Continued Graph the solid boundary line x + y = 70, and shade below it. Graph the solid boundary line 50 + 2x + 3y ≤ 215, and shade below it. The overlapping region is the solution region.

Example 2 Continued Check Test the point (20, 20) in both inequalities. This point represents painting 20 red and 20 gold plates. x + y ≤ 70 50 + 2x + 3y ≤ 215 20 + 20 ≤ 70 50 + 2(20) + 3(20) ≤ 215 40 ≤ 70  150 ≤ 215 

Check It Out! Example 2 Leyla is selling hot dogs and spicy sausages at the fair. She has only 40 buns, so she can sell no more than a total of 40 hot dogs and spicy sausages. Each hot dog sells for $2, and each sausage sells for $2.50. Leyla needs at least $90 in sales to meet her goal. Write and graph a system of inequalities that models this situation.

Check It Out! Example 2 Continued Check Test the point (5, 32) in both inequalities. This point represents selling 5 hot dogs and 32 sausages. 2d + 2.5s ≥ 90 d + s ≤ 40 5 + 32 ≤ 40 2(5) + 2.5(32) ≥ 90 37 ≤ 40  90 ≥ 90 

Example: Maximize/Minimize You are screen-printing T-shirts and sweatshirts to sell at the Polk County Blue Festival. You have at most 20 hours to make shirts. You want to spend no more than $600 on supplies. You want to have at least 50 items to sell. T-shirts take 10 minutes to make and supplies cost $4 each. Sweatshirts take 30 minutes to make and supplies cost $20 each. If you make a profit of $6 for each T-shirt and $6 for each sweatshirt, how many sweatshirts and T-shirts should you make to maximize your profit? What is the maximized profit?

Objective Function: Profit P = 6x +20y Example Continued Let x represent the number of T-Shirts, and let y represent the number of sweatshirts.       Objective Function: Profit P = 6x +20y

Example Continued x  0 y  0 The system of inequalities is x + y ≥ 50 10x +30y ≤ 1200 4x + 20y ≤ 200

What happens to the profit as we move along the boundaries? P = 6x +20y

Linear Programming Find the minimum and maximum values by graphing the inequalities and finding the vertices of the polygon formed. Substitute the vertices into the function and find the largest and smallest values.

A (50,0) P = 6(50) + 20(0) = 300 B (25,25) P = 6(25) + 20(25) = 650 C (75,15) P = 6(75) + 20(15) = 750 D (120,0) P = 6(120) + 20(0) = 720

Homework: pg160 #10,11,13,16,17,19