**Maximizing the Profit of a Business**

Maximizing the Profit of a Business

One of the applications of graphing linear inequalities is in optimization problems. In an optimization

problem, we want to find either the maximum or minimum value of a function over a given region. In

the case of a linear function, and a region given by a system of linear inequalities, any maximum and

minimum values will always occur at one of the vertices of the region. For this project consider the

following situation:

A manufacturer produces two items: computer desks and bookcases. Each item requires processing in

each of two departments. Manufacturing a computer desk requires 2 hours in Department A and

4 hours in Department B. Manufacturing a bookcase requires 3 hours in Department A and 3 hours in

Department B. Each week, Department A has 42 hours available and Department B has 60 hours

available for production. The profits for selling the items are $70 for each desk and $55 for each

bookcase. If all the units can be sold, how many of each should be made to maximize profits?

1. Let be the number of computer desks manufactured and let be the number of bookcases

manufactured. Find a linear inequality (in terms of and ) for the hours used in Department A. Write

this inequality in the box at the bottom of this page. Find a linear inequality (in terms of and ) for the

hours used in Department B. Write this inequality in the box at the bottom of this page.

Keeping in mind that the manufacturer cannot produce a negative number of items, we have two more

inequalities. Write these two inequalities in the box at the bottom of this page:

2. Next, let be the profit from the sale of desks and bookcases. At the bottom of the box, write an

equation for the profit function in terms of and .

The box below should now contain four linear inequalities and one profit function. Together, these

describe the manufacturing situation. This is an example of what is known mathematically as a linear

programming problem.

3. Graph the overlap of all four inequalities on the grid below. Use the horizontal axis to represent

and the vertical axis to represent . Make sure your graph has the overlap shaded. The shaded region

should have four vertices or corners. The points inside the shaded region represent the quantities of

desks and bookcases that can be produced in a week, subject to the time constraints. Points outside the

shaded region represent quantities of desks and bookcases that cannot be produced, because the

departments would not have enough time.

Hints: You may graph the inequalities either by solving them for to get them into slope-intercept form,

or you can leave the inequalities in standard form and graph them by finding the – and -intercepts.

Make sure you graph the lines carefully enough so that you can see where they cross the axes and

where they cross each other.

4. The shaded region should have four vertices. Find the coordinates of each vertex and write

all four of them below. To find the intersection of the two slanted lines, you may want to solve the

2 by 2 system made up of their equations. It might not be a good idea to rely solely on your graph

to find this intersection point.

5. Take the four points you found in step 4 and plug each one into the profit function to complete

the table below. Of these four points, which one results in the largest profit? This point will have

the maximum profit out of any of the points in the shaded region. Lastly, write a sentence to

answer these two questions: How many units of each type of furniture should the manufacturer

build and sell to make the maximum profit? What is the maximum profit?

Vertex

Profit

6. The following month, the company that supplies materials to the manufacturer has lowered the

cost of some of the materials used to make the computer desks. This means that the manufacturer

now makes a profit of $80 on each computer desk produced and sold. If the profit for the

bookcases remains the same, write a new profit function below.

7. Even though the profit has changed, the time required to produce the furniture—and the time

available in the two departments—remains the same. This means that we will still be dealing with

the same shaded region from step 4. Fill in the table below using the new profit function. Write a

sentence to answer these two questions: With the new profit, how many units of each type of

furniture should the manufacturer build and sell to make the maximum profit? Now what is the

maximum profit?

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