3-4 Linear Programming

Transcription

1 Warm Up Determine if the given ordered pair is a solution of x + y 6 1. (3, 3) 2. (10, 1) x 2y >10 LEARNING GOALS FOR LESSON Write systems of equations modeling real-world situations, and graph a feasible region to represent system Solve linear programming problems Determine the Maximum or Minimum value of a feasible region programming is method of finding a maximum or minimum value of a function that satisfies a given set of conditions called constraints. A is one of the inequalities in a linear programming problem. The solution to the set of constraints can be graphed as a. Example: You want to start a business selling your hand made jewelry made of wire and glass beads. What is the main goal (objective) of this business? Will you have to spend any money to start this business? How much could you make? What other things will you have to think about?

2 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. Write the constraints: LG x 0 y 0 The number of batches cannot be negative. OATS FLOUR To optimize the profit or to make the most loaves of bread we would look to the answers

3 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. Example 2: Solving Linear Programming Problems LG 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.

4 Check Yourself! LG Maximize the objective function P = 25x + 30y under the following constraints. x 0 y x + 5y 20 3x + 2y 12

5 Example 3A: Problem-Solving Application LG 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. 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. The number of advanced players is at least 80. The number of intermediate players cannot be negative. There are at least 60 more advanced players than intermediate players. The total cost must be no more than $13,250. Let P = the number of players sent to camp. The objective function is P = x + y. Test the vertices!

6 Example 3B: Problem-Solving Application LG 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? 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. The number of Bookcase A cannot be negative. The number of Bookcase B cannot be negative. There are 8 or less of Bookcase A. There are 12 or less of Bookcase B. The total shelf space is at least 320 feet. Let P = The number of Bookcase A and Bookcase B. The objective function is