Applications of Polynomials

Learning Outcomes

  • Geometric Applications
    • Write a polynomial representing the perimeter of a shape
    • Write a polynomial representing the area of a surface
    • Write a polynomial representing the volume of a solid
  • Cost, Revenue, and Profit Polynomials
    • Write a profit polynomial given revenue and cost polynomials
    • Find profit for given quantities produced

In this section we will explore ways that polynomials are used in applications of perimeter, area, and volume. First, we will see how a polynomial can be used to describe the perimeter of a rectangle.

Example

A rectangular garden has one side with a length of [latex]x+7[/latex] and another with a length [latex]2x + 3[/latex]. Find the perimeter of the garden.

Rectangle with height x+7 and length 2x+3.

In the following video you are shown how to find the perimeter of a triangle whose sides are defined as polynomials.

The area of a circle can be found using the radius of the circle and the constant pi in the formula [latex]A=\pi{r^2}[/latex]. In the next example we will use this formula to find a polynomial that describes the area of an irregular shape.

Example

Find a polynomial for the shaded region of the figure.

circle with middle extracted to form a ring shape. Inner radius labeled as r=3, outer radius labeled as R= r.

In the video that follows, you will be shown an example of determining the area of a rectangle whose sides are defined as polynomials.

 

pi

Pi

A note about pi.

It is easy to confuse pi as a variable because we use a greek letter to represent it.  We use a greek letter instead of a number because nobody has been able to find an end to the number of digits of pi.  To be precise and thorough, we use the greek letter as a way to say: “we are including all the digits of pi without having to write them”. The expression for the area of the shaded region in the example above included both the variable r, which represented an unknown radius and the number pi.  If we needed to use this expression to build a physical object or instruct a machine to cut specific dimensions, we would round pi to an appropriate number of decimal places.

 

In the next example, we will write the area for a rectangle in two different ways, one as the product of two binomials and the other as the sum of four rectangles. Because we are describing the same shape two different ways, we should end up with the same expression no matter what way we define the area.

Example

Write two different polynomials that describe the area of of the figure. For one expression, think of the rectangle as one large figure, and for the other expression, think of the rectangle as the sum of 4 different rectangles.

Rectangle with side length y+9 and y+7

The last example we will provide in this section is one for volume.  The volume of regular solids such as spheres, cylinders, cones and rectangular prisms are known.  We will find an expression for the volume of a cylinder, which is defined as [latex]V=\pi{r^2}h[/latex].

 

Example

Define a polynomial that describes the volume of the cylinder shown in the figure:

Cylinder with height = 7 and radius = (t-2)

In this last video, we present another example of finding the volume of a cylinder whose dimensions include polynomials.

Cost, Revenue, and Profit Polynomials

In the systems of linear equations section, we discussed how a company’s cost and revenue can be modeled with two linear equations. We found that the profit region for a company was 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.

Pile of cell phones

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.

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:

  1. x = 100 phones
  2. x = 25,000 phones
  3. x=60,000 phones

Interpret your results.

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

Summary

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.

In this section we defined polynomials that represent perimeter, area and volume of well-known shapes.  We also introduced some convention about how to use and write [latex]\pi[/latex] when it is combined with other constants and variables. The next application will introduce you to cost and revenue polynomials.  We explored cost and revenue equations in the module on Systems of Linear Equations, now we will see that they can be more than just linear equations, they can be polynomials.