Alg 2 BC - U3Day 4 - Linear Programming
Thinking Skill: Develop Intellectual Perseverance
Getting Started: Your friend Walter has decided to run for state representative. You are responsible for helping him decide how to spend his money in order to win the election.
What kinds of things do you need to consider? What kind of information do you need to gather?
Linear programmingis a technique that identifies the maximum or minimum value of some quantity.
Objective function: ______
Constraints: ______
Feasible region: ______
Example 1: Suppose you go to the store to buy some tapes and CDs. The store only has 7 CDs and 10 tapes available for sale. You want at least 4 CDs and at least 10 hours of recorded music. Each tape holds about 45 minutes of music, and each CD holds about an hour. If tapes cost $8 each and CDs cost $12 each, what is the least amount of money you can spend to get what you want?
Let x represent the number of tapes purchased. Let y represent the number of CDs purchased.
OBJECTIVE FUNCTION:
Write a system of inequalities to model the problem :
Graph your system of inequalities.
Does each ordered pair satisfy the system you graphed?
a. (4,7) b. (12,7) c. (7,6) d. (9,4)e. (10,4)
Vertex principle of linear programming:
If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices in the feasible region.
Example 2:
Find the values of x and y that maximize and minimize P for the objective function . What is the value of P for each vertex?
Constraints:
Example 3:
Suppose you are selling cases of popcorn and pretzels. You can order no more than a total of 500 packages and spend no more than $600. How can you maximize your profit? How much is the maximum profit?
PopcornPretzels
12 packages per case20 packages per case
You pay…. $24 per caseYou pay …. $15 per case
Sell at……. $3.50 per packageSell at ……. $1.50 per package
$18 profit per case$15 profit per case
Example 4:
Paul makes and sells bread. A loaf of Irish soda bread is made with 2 cups of flour and ¼ cup of sugar. A loaf of raisin bread is made with 4 cups of flour and 1 cup of sugar. He will make a profit of $1.50 on each loaf of Irish soda bread and a profit of $4 on each loaf of raisin bread. He has 16 cups of flour and 3 cups of sugar. How many of each kind of bread should Paul make to maximize the profit? What is the maximum profit?
Example 5: (Do on your calculator): Fitzkee’s Candies is selling chocolate covered pretzels and chocolate covered peanut butter cups. The pretzels can be produced at the rate of 40 pounds per hour, and the peanut butter cups at the rate of 65 pounds per hour. Fitzkee’s earns a profit of $5 per pound on the pretzels and $3 per pound on the peanut butter cups. The company has a total of 50 man-hours available. They know they will need a minimum of 900 pounds of peanut butter cups. In addition, they know they will sell at least half as many pretzels as peanut butter cups. Find the number of pounds of pretzels and pounds of peanut butter cups they should sell in order to maximize profit.
Example 6: A farmer has 10 acres to plant wheat and rye. He has to plant at least 7 acres. However, he has only $1200 to spend and each acre of wheat costs $200 to plant and each acre of rye costs $100 to plant. Moreover the farmer has to get the planting done in 12 hours; and it takes an hour to plant an acre of wheat and 2 hours to plant an acre of rye. If the profit is $500 per acre of wheat and $300 per acre of rye, how many acres of each should be planted to maximize profit?
Example 7: 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 each source must be processed each day in order to maximize the amount of gold extracted subject the above constraints?