NEW SEATS! NO COMPLAINING! Homework out (that was two sheets!) 1 st sheet, all of the front, 6 of 8 on the back 2 nd sheet, you try #1 and 3, homework # 2 Big day today! Pick up and think about this warm up

R EVIEW - GRAPHING WITH COVER UP METHOD. 3x + 2y = -6-x + 4y = 6

Linear Programming

What is it????  Linear programming is a technique used to find the maximum or minimum value of an objective equation.

W HAT DOES THAT MEAN ???

Example 1: Find the maximum value for the objective function P = 3x + 2y in the feasible region PointValue What do I want: the biggest or the smallest?

Example 2: Find the minimum value for the objective function P = 5x – 2y in the feasible region. PointValue What do I want: the biggest or the smallest?

TRY IT YOURSELF! M AXIMIZE P = 12 X + 2 Y PointValue What do I want: the biggest or the smallest?

TRY IT YOURSELF #2! M INIMIZE C = 3 X - Y PointValue What do I want: the biggest or the smallest?

Example 3: Find the values of x and y that minimizes the objective function P = 3x + 2y. PointValue What do I want: the biggest or the smallest?

Example 4: Find the values of x and y that maximize the objective function P = 3x + 4y. 2 ≤ x ≤ 6 1 ≤ y ≤ 5 x + y ≤ 8 PointValue What do I want: the biggest or the smallest?

E XAMPLE 5: A FURNITURE COMPANY CAN PRODUCE THE FOLLOWING IN A GIVEN DAY TO 60 TABLES - 30 TO 70 CHAIRS - AT MOST 80 UNITS - PROFIT ON A TABLE IS $150 - PROFIT ON A CHAIR IS $65 H OW MANY TABLES AND CHAIRS SHOULD THE FURNITURE COMPANY MAKE PER DAY TO MAXIMIZE PROFITS ? x = # of tables y = # of chairs Objective Function: P = 150x + 65y Constraints: PointValue

Variables: Objective Function: Constraints: Example 6: A gold processor has two sources of gold ore, source A and source B. In order to keep his plant running, at least three tons of ore must be processed each day. Ore from source A costs $20 per ton to process, and ore from source B costs $10 per ton to process. Costs must be kept to less than $80 per day. Moreover, Federal Regulations require that the amount of ore from source B cannot exceed twice the amount of ore from source A. If ore from source A yields 2 oz. of gold per ton, and ore from source B yields 3 oz. of gold per ton, how many tons of ore from both sources must be processed each day to maximize the amount of gold extracted subject to the above constraints?

Objective Function: Constraints: BONUS: A CANDY MAKER MUST MIX CANDY FOR THE V ALENTINE ’ S D AY HOLIDAY IN AN ATTEMPT TO MAXIMIZE PROFIT. R EESE ’ S P EANUT B UTTER CUPS PROFIT $1.75 A POUND AND H ERSHEY ’ S K ISSES IS $1.50 PER POUND. H OW MUCH OF EACH CANDY SHOULD SHE MIX TO MAXIMIZE THE PROFIT FOLLOWING THE GIVEN CRITERIA ? AT MOST 17 POUNDS OF CANDY ONLY 9 POUNDS OF REESES ’ S PEANUT BUTTER CUPS 12 CUPS OF HERSHEY ’ S KISSES AT LEAST 5 POUNDS OF R EESE ’ S P EANUT B UTTER CUPS AT LEAST 8 POUNDS OF H ERSHEY ’ S K ISSES

~ CLASS WORK ~ Attempt the bonus with a partner ;)

~ HOME WORK ~ Worksheet! #1-2

) Find the maximal and minimal values given P = 3x + 4y.

F OR REAL … IS THERE AN EASIER WAY ? A furniture company can produce the following in a given day. 30 to 60 tables 40 to 100 chairs at most 120 units profit on a table is $150 profit on a chair is $65 How many tables and chairs should the furniture company make per day to maximize profits? What profits the most? That’s where we start!

A SK YOURSELF … Do tables or chairs make the most profit? How many of those can I make? Can I make anything else? Am I fitting within all my given criteria?

# tables # chairs Profit! 60 $12,900 Some might prefer a table…

E X 1. A calculator company produces a scientific calculator and a graphing calculator. Long-term projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can be made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a $2 loss, but each graphing calculator produces a $5 profit, how many of each type should be made daily to maximize net profits?

1 st- Summarize exactly what we know so I don’t’ have to keep searching through those paragraphs! 2 nd - Follow the steps above. 3 rd - Answer the question

F OR THE TEACHER ONLY R = –2 x + 5 y, subject to: 100 – x When you test the corner points at (100, 170), (200, 170), (200, 80), (120, 80), and (100, 100), you should obtain the maximum value of R = 650 at ( x, y ) = (100, 170). That is, the solution is "100 scientific calculators and 170 graphing calculators".

R ELEASED TEST QUESTION ! Amber sells lemonade and limeade. Making lemonade takes 1 cup sugar and 12 lemons Making limeade takes 0.5 cups sugar and 10 lemons Amber has 7 cups of sugar and 100 lemons She makes $2 profit on lemonade She makes $1.50 profit on limeade How many batches of limeade should Amber make to maximize her profits?

Y OU TRY ! A TOTALLY MADE UP PROBLEM ! Assume there are only two foods, steak and cereal, and two nutrients, protein and iron, that people need to survive. Each day, a person must consume at least 60 units of iron and at least 70 units of protein One unit of cereal cost $2 and contains 30 units of iron and 5 units of protein One unit of steak costs $20 and contains 15 units of iron and 10 units of protein Find the cheapest diet which will satisfy the minimum daily requirements.

F OR THE TEACH Only cereal

R EVIEW FOR Q UIZ … Worksheet #3-4

S URPRISE ! P ARTNER Q UIZ. G ET WITH THIS PERSON !

Q UIZ ! Must finish #1 before leaving today. Will have 15 minutes to do #2 at the very beginning of class tomorrow. Plan on not coming tomorrow? Looks like you’re quizzing along bucko