Models and Applications

Learning Objectives

In this section, you will:

• Set up a linear equation to solve a real-world application.
• Use a formula to solve a real-world application.

Figure 1 Credit: Kevin Dooley

Neka is hoping to get an A in his college algebra class. He has scores of 75, 82, 95, 91, and 94 on his first five tests. Only the final exam remains, and the maximum number of points that can be earned is 100. Is it possible for Neka to end the course with an A? A simple linear equation will give Neka his answer.

Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce x widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.

Setting up a Linear Equation to Solve a Real-World Application

To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write 0.10x.0.10𝑥. This expression represents a variable cost because it changes according to the number of miles driven.

If a quantity is independent of a variable, we usually just add or subtract it, according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost C.𝐶.

C=0.10x+50𝐶=0.10𝑥+50

When dealing with real-world applications, there are certain expressions that we can translate directly into math. Table 1 lists some common verbal expressions and their equivalent mathematical expressions.

Verbal	Translation to Math Operations
One number exceeds another by a	x,x+a𝑥,𝑥+𝑎
Twice a number	2×2𝑥
One number is a more than another number	x,x+a𝑥,𝑥+𝑎
One number is a less than twice another number	x,2x−a𝑥,2𝑥−𝑎
The product of a number and a, decreased by b	ax−b𝑎𝑥−𝑏
The quotient of a number and the number plus a is three times the number	xx+a=3x𝑥𝑥+𝑎=3𝑥
The product of three times a number and the number decreased by b is c	3x(x−b)=c3𝑥(𝑥−𝑏)=𝑐

HOW TO

Given a real-world problem, model a linear equation to fit it.

1. Identify known quantities.
2. Assign a variable to represent the unknown quantity.
3. If there is more than one unknown quantity, find a way to write the second unknown in terms of the first.
4. Write an equation interpreting the words as mathematical operations.
5. Solve the equation. Be sure the solution can be explained in words, including the units of measure.

EXAMPLE 1

Modeling a Linear Equation to Solve an Unknown Number Problem

Find a linear equation to solve for the following unknown quantities: One number exceeds another number by 1717 and their sum is 31.31. Find the two numbers.

TRY IT #1

Find a linear equation to solve for the following unknown quantities: One number is three more than twice another number. If the sum of the two numbers is 36,36, find the numbers.

EXAMPLE 2

Setting Up a Linear Equation to Solve a Real-World Application

There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05/min talk-time. Company B charges a monthly service fee of $40 plus $.04/min talk-time.

1. ⓐWrite a linear equation that models the packages offered by both companies.
2. ⓑ If the average number of minutes used each month is 1,160, which company offers the better plan?
3. ⓒIf the average number of minutes used each month is 420, which company offers the better plan?
4. ⓓHow many minutes of talk-time would yield equal monthly statements from both companies?

TRY IT #2

Find a linear equation to model this real-world application: It costs ABC electronics company $2.50 per unit to produce a part used in a popular brand of desktop computers. The company has monthly operating expenses of $350 for utilities and $3,300 for salaries. What are the company’s monthly expenses?

Using a Formula to Solve a Real-World Application

Many applications are solved using known formulas. The problem is stated, a formula is identified, the known quantities are substituted into the formula, the equation is solved for the unknown, and the problem’s question is answered. Typically, these problems involve two equations representing two trips, two investments, two areas, and so on. Examples of formulas include the area of a rectangular region, A=LW;𝐴=𝐿𝑊; the perimeter of a rectangle, P=2L+2W;𝑃=2𝐿+2𝑊; and the volume of a rectangular solid, V=LWH.𝑉=𝐿𝑊𝐻. When there are two unknowns, we find a way to write one in terms of the other because we can solve for only one variable at a time.

EXAMPLE 3

Solving an Application Using a Formula

It takes Andrew 30 min to drive to work in the morning. He drives home using the same route, but it takes 10 min longer, and he averages 10 mi/h less than in the morning. How far does Andrew drive to work?

Analysis

Note that we could have cleared the fractions in the equation by multiplying both sides of the equation by the LCD to solve for r.𝑟.

r(12)6×r(12)3r3r−rr======(r−10)(23)6×(r−10)(23)4(r−10)4r−40−4040𝑟(12)=(𝑟−10)(23)6×𝑟(12)=6×(𝑟−10)(23)3𝑟=4(𝑟−10)3𝑟=4𝑟−40−𝑟=−40𝑟=40

TRY IT #3

On Saturday morning, it took Jennifer 3.6 h to drive to her mother’s house for the weekend. On Sunday evening, due to heavy traffic, it took Jennifer 4 h to return home. Her speed was 5 mi/h slower on Sunday than on Saturday. What was her speed on Sunday?

EXAMPLE 4

Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is 5454 ft. The length is 33 ft greater than the width. What are the dimensions of the patio?

TRY IT #4

Find the dimensions of a rectangle given that the perimeter is 110110 cm and the length is 1 cm more than twice the width.

EXAMPLE 5

Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in. The length is 66 in. more than the width. Find the area of the graph paper.

TRY IT #5

A game room has a perimeter of 70 ft. The length is five more than twice the width. How many ft² of new carpeting should be ordered?

EXAMPLE 6

Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is 88 inches, and the volume is 1,600 in.³.

Analysis

Note that the square root of W2𝑊2 would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.