Graph the Equations and Inequalities on Another Sheet of Paper

Algebra 2 Name ______

Linear Programming

Assignment 1

Graph the equations and inequalities on another sheet of paper.

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Assignment 2

Solve on another sheet of paper.

1.  A ski manufacturer makes two types of skis and has a fabricating department and a finishing department. A pair of downhill skis requires 6 hours to fabricate and 1 hour to finish. A pair of cross-country skis requires 4 hours to fabricate and 1 hour to finish. The fabricating department has 108 hours of labor available per day. The finishing department has 24 hours of labor available per day. The company makes a profit of $40 on each pair of downhill skis and a profit of $30 on each pair of cross-country skis.

a.  Write a system of linear inequalities to represent the constraints.

b.  Write the objective function for the profit.

2.  A school dietician wants to prepare a meal of meat and vegetables that has the lowest possible fat and that meets the Food and Drug Administration recommended daily allowances (RDA) of iron and protein. The RDA minimums are 20 milligrams of iron and 45 grams of protein. Each 3-ounce serving of meat contains 45 grams of protein, 10 milligrams of iron, and 4 grams of fat. Each 1-cup serving of vegetables contains 9 grams of protein, 6 milligrams of iron, and 2 grams of fat. Let x represent the number of 3-ounce servings of meat, and let y represent the number of 1-cup servings of vegetables.

a.  Write a system of linear inequalities to represent the constraints.

b.  Write the objective function for the number of grams of fat.

3.  A farmer has 90 acres available for planting millet and alfalfa. Seed costs $4 per acre for millet and $6 per acre for alfalfa. Labor costs are $20 per acre for millet and $10 per acre for alfalfa. The expected income is $110 per acre for millet and $150 per acre for alfalfa. The farmer intends to spend no more than $480 for seed and $1400 for labor.

a.  Write a system of linear inequalities to represent the constraints.

b.  Write the objective function that maximizes the income.

4.  Graph the system

5.  Graph the system

Assignment 3

Find the vertices of the feasible regions from Assignment 2 (all 5 problems)

Show your work on another sheet of paper.

Assignment 4

Use your work from Assignments 2 and 3 to solve problems 1 – 3 in Assignment 2.

In problem 1, find the maximum profit for the given constraints.

In problem 2, find the minimum number of grams of fat for the given constraints.

In problem 3, find the maximum income for the given constraints.

Also solve the following new problems.

A.  Find the maximum and minimum values, if they exist, of the objective function given the set of constraints: .

B.  A carpenter makes bookcases in two sizes, large and small. It takes 6 hours to make a large bookcase and 2 hours to make a small one. The profit on a large bookcase is $50, and the profit on a small bookcase is $20. The carpenter can spend only 24 hours per week making bookcases, and must make at least 2 of each size per week.

i.  Write a system of linear inequalities to represent the constraints.

ii.  Graph the feasible region.

iii.  Write the objective function for the profit, and find the maximum profit for the given constraints.