Polynomial Applications Involving Cost, Revenue, and Profit

Learning Objectives

Write a profit polynomial given revenue and cost polynomials
Find profit for given quantities produced

A company’s cost and revenue can be modeled with two linear equations. The profit region for a company is the area between the two lines where the company would make money based on how much was produced. In this section, we will see that sometimes polynomials are used to describe cost and revenue.

Profit is typically defined in business as the difference between the amount of money earned (revenue) by producing a certain number of items and the amount of money it takes to produce that number of items. When you are in business, you definitely want to see profit, so it is important to know what your cost and revenue is.

Cell Phones

For example, let’s say that the cost to a manufacturer to produce a certain number of things is C and the revenue generated by selling those things is R. The profit, P, can then be defined as

P = R-C

The example we will work with is a hypothetical cell phone manufacturer whose cost to manufacture x number of phones is [latex]C=2000x+750,000[/latex], and the Revenue generated from manufacturing x number of cell phones is [latex]R=-0.09x^2+7000x[/latex].

Example

Define a Profit polynomial for the hypothetical cell phone manufacturer.

Show Solution

Mathematical models are great when you use them to learn important information. The cell phone manufacturing company can use the profit equation to find out how much profit they will make given x number of phones are manufactured. In the next example, we will explore some profit values for this company.

Example

Given the following numbers of cell phones manufactured, find the profit for the cell phone manufacturer:

x = 100 phones
x = 25,000 phones
x=60,000 phones

Interpret your results.

Show Solution

Read and Understand: The profit polynomial defined in the previous example,[latex]P=-0.09x^2+5000x-750,000[/latex], gives profit based on x number of phones manufactured. We need to substitute the given numbers of phones manufactured into the equation, then try to understand what our answer means in terms of profit and number of phones manufactured.

We will move straight into write and solve since we already have our polynomial. It is probably easiest to use a calculator since the numbers in this problem are so large.

Write and Solve:

Substitute x = 100

[latex]\begin{array}{c}P=-0.09x^2+5000x-750,000\\=-0.09\left(100\right)^2+5000\left(100\right)-750,000\\=-900+500,000-750,000\\=-250,900\end{array}[/latex]

Interpret: When the number of phones manufactured is 100, the profit for the business is $-250,000. This is not what we want! The company must produce more than 100 phones to make a profit.

Write and Solve:

Substitute x = 25,000

[latex]\begin{array}{c}P=-0.09x^2+5000x-750,000\\=-0.09\left(25000\right)^2+5000\left(25000\right)-750,000\\=-6120000+125,000,000-750,000\\=118,130,000\end{array}[/latex]

Interpret: When the number of phones manufactured is 25,000, the profit for the business is $118,130,000. This is more like it! If the company makes 25,000 phones it will make a profit after it pays all it’s bills.

If this is true, then the company should make even more phones so it can make even more money, right? Actually, something different happens as the number of items manufactured increases without bound.

Write and Solve:

Substitute x = 60,000

[latex]\begin{array}{c}P=-0.09x^2+5000x-750,000\\=-0.09\left(60000\right)^2+5000\left(60000\right)-750,000\\=-324,000,000+300,000,000-750,000\\=-24,750,000\end{array}[/latex]

Interpret: When the number of phones manufactured is 60,000, the profit for the business is $-24,750,000. Wait a minute! If the company makes 60,000 phones it will lose money, what happened? At some point, the cost to manufacture the phones will overcome the amount of profit that the business can make. If this is interesting to you, you may enjoy reading about Economics and Business models.

In the video that follows, we present another example of finding a polynomial profit equation.

We have shown that profit can be modeled with a polynomial, and that the profit a company can make based on a business model like this has it’s bounds.

Objective 3