Write polynomials involving perimeter, area, and volume
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.
Show Solution
The perimeter of a rectangle is the sum of its side lengths.
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 area of the shaded region of the figure.
Show Solution
Read and Understand: We need to describe the area of the shaded region of this shape using polynomials. We know the formula for the area of a circle is [latex]A=\pi{r^2}[/latex]. The figure we are working with is a circle with a smaller circle cut out.
Define and Translate: The larger circle has radius = r, and the smaller circle has radius= [latex]3[/latex]. If we find the area of the larger circle, then subtract the area of the smaller circle, the remaining area will be the shaded region. First define the area of the larger circle:
[latex]A_{1}=\pi{r^2}[/latex]
Then define the area of the smaller circle.
[latex]A_{2}=\pi{3^2}=9\pi[/latex]
Write and Solve: The shaded region can be found by subtracting the smaller area from the larger area.
The area of the shaded region is [latex]\pi{r^2}-9\pi[/latex]
Answer
[latex]\pi{r^2}-9\pi[/latex]
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
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 [latex]4[/latex] different rectangles.
Show Solution
First, we will define the polynomial that describes the area of the rectangle as one figure.
Read and Understand: We are tasked with writing an expressions for the area of the figure above. The area of a rectangle is given as [latex]A=lw[/latex]. We need to consider the whole figure in our dimensions.
Define and Translate: Use the formula for area: [latex]A=lw[/latex], the sides of the figure are the sum of the defined sides. [latex]\begin{array}{c}l=\left(y+7\right)\\w=\left(y+9\right)\end{array}[/latex]
You could define [latex]\begin{array}{c}w=\left(y+7\right)\\l=\left(y+9\right)\end{array}[/latex] because it doesn’t matter in which order you multiply.
Now we will find an expression for the area of the whole figure as comprised by the areas of the four rectangles added together.
Read and Understand: The area of a rectangle is given as [latex]A=lw[/latex]. We need to first define the areas of each rectangle, then sum them all together to get the area of the whole figure. It helps to label the four rectangle in the figure so you can keep the dimensions organized.
Define and Translate: Use the formula for area: [latex]A=lw[/latex] for each rectangle:
[latex]A_{1}=7\cdot{y}[/latex]
[latex]A_{2}=7\cdot{9}=63[/latex]
[latex]A_{3}=y\cdot{y}=y^2[/latex]
[latex]A_{4}=y\cdot{9}[/latex]
Write and Solve:
[latex]\begin{array}{c}A=A_{1}+A_{2}+A_{3}+A_{4}\\=7y+63+y^2+9y\\\text{ reorganize and simplify }\\=y^2+16y+63\end{array}[/latex]
Answer 2
[latex]A=y^2+16y+63[/latex]
Hopefully, it isn’t surprising that both expressions simplify to the same thing.
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:
Show Solution
Read and Understand: We are tasked with writing an expressions for the volume of the cylinder in the figure above. The volume of a cylinder is given as[latex]V=\pi{r^2}h[/latex], where [latex]\pi[/latex] is a constant, and r is the radius and h is the height of the cylinder.
Define and Translate: Use the formula for volume:[latex]V=\pi{r^2}h[/latex], we need to define r and h.
Note that we usually write other constants that are multiplied by [latex]\pi[/latex] in front of it. We can now distribute [latex]7\pi[/latex] to each term in the polynomial.
Note again how we left [latex]\pi[/latex] as a greek letter. If we needed to use this calculation for measurement of materials, we would round pi, or a computer would round for us.
In this last video, we present another example of finding the volume of a cylinder whose dimensions include polynomials.
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. Next we will see that cost and revenue equations can be polynomials.
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Polynomial Multiplication Application - Volume of a Cylinder. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/g-g_nSsfGs4. License: CC BY: Attribution
Revision and Adaptation. Provided by: Lumen Learning. License: CC BY: Attribution
Unit 11: Exponents and Polynomials, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: . License: CC BY: Attribution
