Pre-AP Algebra 2 Lesson 2-6 Linear Programming Problems

Transcription

1 Lesson 2-6 Linear Programming Problems Objectives: The students will be able to: use sstems of linear inequalities to solve real world problems. set up constraints & objective functions for linear programming problems. Materials: Hw #2-5 answers overhead; tall sheets; Bellringer handout and answers overhead; note-taking templates; pair work; homework #2-6 Time Activit 5 min Review Homework Show the answers to #2-5 on the overhead. Students correct their answers. Pass around a tall sheet. 10 min Homework Presentations Review the top 2 or 3 problems. 20 min Do Now Too man inequalities! Students graph sstems of 3 or more inequalities. 25 min Direct Instruction Background Information: To graph a sstem of inequalities: 1) Graph each inequalit 2) The solution of the sstem is the area shaded b all inequalities Concepts: Sstems of inequalities can be used for real-life problems. Eample: A potter wants to make and sell serving bowls and plates. A bowl uses 5 pounds of cla. A plate uses 4 pounds of cla. The potter has 40 pounds of cla and wants to make at least 4 bowls. The profit on a bowl is $35 and the profit on a plate is $30. How man bowls and how man plates should the potter make in order to maimize profit? 1) Use the information given to write down the constraints. 2) Graph the constraints 3) Find the points of intersection of the feasible region 4) Write an equation for total profit (this is the objective function) 5) Evaluate the objective function at each verte. Eample: Suppose a farmer has 150 acres available for planting corn and cotton. The cotton seeds cost $3 per acre and the corn seeds cost $5 per acre. The total labor costs for cotton will be $15 per acre and the total labor costs for corn will be $8 per acre. The farmer epects the income from cotton to be $80 per acre and the income from the corn to be $110 per acre. The farmer can spend no more than $540 on seeds and $1800 on labor. How much corn and cotton should the farmer plant in order to maimize his income? 20 min Pair Work Hand out the Solving Linear Programming Practice sheet for students to work on. Homework #2-6: Linear Programming

2 Lesson 2-6 Bellringer Graph each sstem of linear inequalities 1) Too man inequalities! 2) 3)

3 Lesson 2-6 Notes Background Information: To graph a sstem of inequalities: 1) Graph each inequalit 2) The solution of the sstem is the area shaded b all inequalities Concepts: Sstems of inequalities can be used for real-life problems. Eample: A potter wants to make and sell serving bowls and plates. A bowl uses 5 pounds of cla. A plate uses 4 pounds of cla. The potter has 40 pounds of cla and wants to make at least 4 bowls. The profit on a bowl is $35 and the profit on a plate is $30. How man bowls and how man plates should the potter make in order to maimize profit? 1) Use the information given to write down the constraints. 2) Graph the constraints 3) Find the points of intersection of the feasible region 4) Write an equation for total profit (this is the objective function) 5) Evaluate the objective function at each verte.

4 Lesson 2-6 Pairwork Solving Linear Programming Problems 1. Trees in urban areas help keep air fresh b absorbing carbon dioide. A cit has $2100 to spend on planting spruce and maple trees. The land available for planting is 45,000 square feet. Spruce trees cost $30 to plant and require 600 square feet of space. Maple trees cost $40 to plant and require 900 square feet of space. Spruce trees absorb 650 lb/r of carbon dioide and maple trees absorb 300 lb/r of carbon dioide. How man of each tree should the cit plant to maimize carbon dioide absorption? 2. A to manufacturer wants to minimize her cost for producing two lines of to airplanes. Because of the suppl of materials, no more than 40 Fling Bats can be built each da, and no more than 60 Fling Falcons can be built each da. There are enough workers to build at least 70 to airplanes each da. It costs $12 to manufacture a Fling Bat and $8 to build a Fling Falcon. What is the minimum possible cost each da?

5 Lesson 2-6 Pairwork 3. A seafood restaurant owner orders at least 50 fish. He cannot use more than 30 amberjack or more than 35 flounder. Amberjack costs $4 each and flounder costs $3 each. How man of each fish should he use to minimize his cost? 4. Juan makes two tpes of wood clocks to sell at local stores. It takes him 2 hours to assemble a pine clock, which requires 1 oz of varnish. It takes 2 hours to assemble an oak clock, which takes 4 oz. of varnish. Juan has 16 oz. of varnish in stock, and can work 20 hours. If he makes $3 profit on each pine clock and $4 on each oak clock, how man of each tpe should he make to maimize his profits?

6 Lesson 2-6 Homework Homework #2-6: Linear Programming Do all work on binder paper (stapled to the back). 1) Contracting: The BJ Electrical Compan needs to hire master electricians and apprentices for a one week project. Master electricians receive a salar of $750 per week and apprentices receive $350 per week. As part of its contract, the compan has agreed to hire at least 30 workers. The local Building Safet Council recommends that each master electrician spend three hours for inspection time during the project. This project should require 25 hours of inspection time. How man of each tpe of worker should be hired to accomplish the project and still meet the contract safet requirements? 2) Marketing: Yumm Ice Cream conducted a surve and found that people liked their black walnut flavor three times more than their tutti-frutti flavor. One distributor wants to order at least 20,000 gallons of the tutti-frutti flavor. The compan has all of the ingredients to produce both flavors, but it has onl 45,000 gallon-size containers available. If each gallon of ice cream sells for $2.95, how man gallons of each tpe flavor should the compan produce? 3) Manufacturing: The Cruiser Biccle compan makes two stles of biccles: the Traveler, which sells for $200, and the Tourister, which sells for $600. Each biccle has the same frames and tires, but the assembl and painting time required for the Traveler is onl one hour, while it takes three hours for the Tourister. There are 300 frames and 360 hours of labor available for production. How man of each model should be produced to maimize revenue? 4) Manufacturing: The Swing-Well Compan produces two tpes of golf clubs: the Driver, which sells for $30, and the Master, which sells for $40. Swing-Well has more orders for the upcoming month than it is capable of producing. Using the production schedule below, what is the maimum revenue that Swing-Well should anticipate for the upcoming month? Process Driver Master Time Available Cutting 2 min 2 min 166 2/3 h Assembl 1 min 3 min 150 h Finishing 2 min 3 min 200 h

7 HW #2-5 1) a. es b. no c. no d. es 2) Graph 3) 4) a. es b. es c. no d. no 5) Graph each inequalit. Pa attention to the tpe of boundar line ou need. a) > 2 b) < 4 6) Determine if each point is a solution to the sstem of inequalities shown. (1, 1) no (4, 3) es (-3, 4) no (1, -5) es (3, 5) no (-5, -1) es 7) 8) 9) Graph these non-linear inequalities: ( 3)

8 10) Graph each sstem of inequalities. Label the solution region(s) with an S S Ignore

9 HW #2-5Tall Sheet 1) a. b. c. d. 2) Graph 3) 4) a. b. c. d. 5) a. b. 6) 7) 8) 9) a. b. 10) a. b. c. d. e. f.

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