3 – 4Linear

Programming Word Problems

Linear Programming Steps

1. Define the variables.2. Write a system of inequalities.3. Graph the system of inequalities.4. Find the coordinates of the vertices of the

feasible region.Sometimes you may have to solve a system of equations to identify the exact point at which two lines intersect.

5. Write the function to be maximized. 6. Substitute the coordinates of the vertices into

the function.7. Select the greatest or least amount. Answer

the problem.

A landscaping company has crews who mow lawns and prune shrubbery. The company schedules 1 hour for mowing jobs and 3 hours for pruning jobs. Each crew is scheduled for no more than 2 pruning jobs per day. Each crew’s schedule is set up for a maximum of 9 hours per day. On the average, the charge for mowing a lawn is $40 and the charge for pruning shrubbery is $120. Find a combination of mowing lawns and pruning shrubs that will maximize the income the company receives per day from one of its crews.

1. Define the variables.

Ex 5

2. Write a system of inequalities.

Since the number of jobs cannot be negative, m and p must be nonnegative numbers.

Mowing jobs take 1 hour. Pruning jobs take 3 hours. There are 9 hours to do the jobs.

There are no more than 2 pruning jobs a day.

3. Graph the system of inequalities.Math Composer 1. 1. 5http: / /www. mathcomposer. com

-3 -2 -1 1 2 3 4 5 6 7 8 9 10

-3

-2

-1

1

2

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9

10

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y

4. Find the coordinates of the vertices of thefeasible region.

From the graph, the vertices are at

5. Write the function to be maximized.

The function that describes the income is

We want to find the maximum value for this function.

6. Substitute the coordinates of the vertices intothe function.

7. Select the greatest amount and answer the question.

(m, p) f(m, p)

A landscaping company has crews who rake leaves and mulch. The company schedules 2 hours for mulching jobs and 4 hours for raking jobs. Each crew is scheduled for no more than 2 raking jobs per day. Each crew’s schedule is set up for a maximum of 8 hours per day. On the average, the charge for raking a lawn is $50 and the charge for mulching is $30. Find a combination of raking leaves and mulching that will maximize the income the company receives per day from one of its crews.

Ex 6

Math Composer 1. 1. 5http: / /www. mathcomposer. com

-7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7

-7

-6

-5

-4

-3

-2

-1

1

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x

y