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

[Pages:11]Pre-AP Algebra 2 Lesson 2-6 ? Linear Programming Problems

Objectives: The students will be able to: use systems of linear inequalities to solve real world problems. set up constraints & objective functions for linear programming problems.

Materials: Hw #2-5 answers overhead; tally sheets; Bellringer handout and answers overhead; note-taking templates; pair work; homework #2-6

Time 5 min

10 min

Activity Review Homework Show the answers to #2-5 on the overhead. Students correct their answers. Pass around a tally sheet. Homework Presentations Review the top 2 or 3 problems.

20 min

Do Now Too many inequalities! Students graph systems of 3 or more inequalities.

25 min Direct Instruction

Background Information: To graph a system of inequalities:

1) Graph each inequality 2) The solution of the system is the area shaded by all inequalities

Concepts: Systems of inequalities can be used for real-life problems.

Example: A potter wants to make and sell serving bowls and plates. A bowl uses 5 pounds of clay. A plate uses 4 pounds of clay. The potter has 40 pounds of clay and wants to make at least 4 bowls. The profit on a bowl is $35 and the profit on a plate is $30. How many bowls and how many plates should the potter make in order to maximize 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 vertex.

20 min

Example: 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 expects 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 maximize his income?

Pair Work Hand out the Solving Linear Programming Practice sheet for students to work on.

Homework #2-6: Linear Programming

Pre-AP Algebra 2 Lesson 2-6 ? Bellringer

Too many inequalities!

Graph each system of linear inequalities 1)

2)

3)

Pre-AP Algebra 2 Lesson 2-6 ? Notes Background Information: To graph a system of inequalities:

1) Graph each inequality 2) The solution of the system is the area shaded by all inequalities Concepts: Systems of inequalities can be used for real-life problems. Example: A potter wants to make and sell serving bowls and plates. A bowl uses 5 pounds of clay. A plate uses 4 pounds of clay. The potter has 40 pounds of clay and wants to make at least 4 bowls. The profit on a bowl is $35 and the profit on a plate is $30. How many bowls and how many plates should the potter make in order to maximize 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 vertex.

Pre-AP Algebra 2 Lesson 2-6 ? Pairwork

Solving Linear Programming Problems

1. Trees in urban areas help keep air fresh by absorbing carbon dioxide. A city 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/yr of carbon dioxide and maple trees absorb 300 lb/yr of carbon dioxide. How many of each tree should the city plant to maximize carbon dioxide absorption?

2. A toy manufacturer wants to minimize her cost for producing two lines of toy airplanes. Because of the supply of materials, no more than 40 Flying Bats can be built each day, and no more than 60 Flying Falcons can be built each day. There are enough workers to build at least 70 toy airplanes each day. It costs $12 to manufacture a Flying Bat and $8 to build a Flying Falcon. What is the minimum possible cost each day?

Pre-AP Algebra 2 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 many of each fish should he use to minimize his cost?

4. Juan makes two types 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 many of each type should he make to maximize his profits?

Pre-AP Algebra 2 Lesson 2-6 ? Homework

Homework #2-6: Linear Programming

Do all work on binder paper (stapled to the back).

1) Contracting: The BJ Electrical Company needs to hire master electricians and apprentices for a one week project. Master electricians receive a salary of $750 per week and apprentices receive $350 per week. As part of its contract, the company has agreed to hire at least 30 workers. The local Building Safety 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 many of each type of worker should be hired to accomplish the project and still meet the contract safety requirements?

2) Marketing: Yummy Ice Cream conducted a survey 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 company has all of the ingredients to produce both flavors, but it has only 45,000 gallon-size containers available. If each gallon of ice cream sells for $2.95, how many gallons of each type flavor should the company produce?

3) Manufacturing: The Cruiser Bicycle company makes two styles of bicycles: the Traveler, which sells for $200, and the Tourister, which sells for $600. Each bicycle has the same frames and tires, but the assembly and painting time required for the Traveler is only one hour, while it takes three hours for the Tourister. There are 300 frames and 360 hours of labor available for production. How many of each model should be produced to maximize revenue?

4) Manufacturing: The Swing-Well Company produces two types 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 maximum revenue that Swing-Well should anticipate for the upcoming month?

Process

Cutting Assembly Finishing

Driver

2 min 1 min 2 min

Master

2 min 3 min 3 min

Time Available

166 2/3 h 150 h 200 h

HW #2-5

1) a. yes 2) Graph

b. no

c. no

d. yes

3)

4) a. yes

b. yes

c. no

d. no

5) Graph each inequality. Pay attention to the type of boundary line you need.

a) |x| > 2

b) |y| < 4

y y

x

x

6) Determine if each point is a solution to the system of inequalities shown.

(1, 1) no

(4, 3)

yes

(-3, 4) no

(1, -5)

yes

(3, 5) no

(-5, -1)

yes

7)

8)

9) Graph these non-linear inequalities: y (x 3)2 3

y

y 1|x 2| 3 2

y

x

x

10) Graph each system of inequalities. Label the solution region(s) with an "S".

y2 y2x y x2

3x 2 y 5 2x 3y 6

y y

S

Ignore

x x

|y| 1 |x| 1

y

x

|y| 1 |x| 1

y

x

y x2 4 y2

y

x

y | x 1| 2 y 1x 4

2

y

x

 ................
................

In order to avoid copyright disputes, this page is only a partial summary.

Google Online Preview   Download