## 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

Josh 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 of points that can be earned is 100. Is it possible for Josh to end the course with an A? A simple linear equation will give Josh 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.

### 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,$\,A=LW;$the perimeter of a rectangle,$\,P=2L+2W;$and the volume of a rectangular solid,$\,V=LWH.\,$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.

### 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$\,r.$

$\begin{array}{ccc}\hfill r\left(\frac{1}{2}\right)& =& \left(r-10\right)\left(\frac{2}{3}\right)\hfill \\ \hfill 6\text{ }×\text{ }r\left(\frac{1}{2}\right)& =& 6\text{ }×\text{ }\left(r-10\right)\left(\frac{2}{3}\right)\hfill \\ \hfill 3r& =& 4\left(r-10\right)\hfill \\ \hfill 3r& =& 4r-40\hfill \\ \hfill -r& =& -40\hfill \\ \hfill r& =& 40\hfill \end{array}$

### Try It

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?

### Solving a Perimeter Problem

The perimeter of a rectangular outdoor patio is$\,54\,$ft. The length is$\,3\,$ft greater than the width. What are the dimensions of the patio?

### Try It

Find the dimensions of a rectangle given that the perimeter is$\,110\,$cm and the length is 1 cm more than twice the width.

### Solving an Area Problem

The perimeter of a tablet of graph paper is 48 in. The length is$\,6\,$in. more than the width. Find the area of the graph paper.

### Try It

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?

### Solving a Volume Problem

Find the dimensions of a shipping box given that the length is twice the width, the height is$\,8\,$inches, and the volume is 1,600 in.3.

#### Analysis

Note that the square root of$\,{W}^{2}\,$would result in a positive and a negative value. However, because we are describing width, we can use only the positive result.

Access these online resources for additional instruction and practice with models and applications of linear equations.

### Key Concepts

• A linear equation can be used to solve for an unknown in a number problem. See (Figure).
• Applications can be written as mathematical problems by identifying known quantities and assigning a variable to unknown quantities. See (Figure).
• There are many known formulas that can be used to solve applications. Distance problems, for example, are solved using the$\,d=rt\,$formula. See (Figure).
• Many geometry problems are solved using the perimeter formula$\,P=2L+2W,$the area formula$\,A=LW,$or the volume formula$\,V=LWH.\,$See (Figure), (Figure), and (Figure).

### Section Exercises

#### Verbal

To set up a model linear equation to fit real-world applications, what should always be the first step?

Use your own words to describe this equation where n is a number:

$5\left(n+3\right)=2n$

If the total amount of money you had to invest was $2,000 and you deposit$\,x\,$amount in one investment, how can you represent the remaining amount? If a man sawed a 10-ft board into two sections and one section was$\,n\,$ft long, how long would the other section be in terms of$\,n$? If Bill was traveling$\,v\,$mi/h, how would you represent Daemon’s speed if he was traveling 10 mi/h faster? #### Real-World Applications For the following exercises, use the information to find a linear algebraic equation model to use to answer the question being asked. Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113? Beth and Ann are joking that their combined ages equal Sam’s age. If Beth is twice Ann’s age and Sam is 69 yr old, what are Beth and Ann’s ages? Ben originally filled out 8 more applications than Henry. Then each boy filled out 3 additional applications, bringing the total to 28. How many applications did each boy originally fill out? For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of$20 and charges of $.05/min for calls. Company B has a monthly fee of$5 and charges $.10/min for calls. Find the model of the total cost of Company A’s plan, using$\,m\,$for the minutes. Find the model of the total cost of Company B’s plan, using$\,m\,$for the minutes. Find out how many minutes of calling would make the two plans equal. If the person makes a monthly average of 200 min of calls, which plan should for the person choose? For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan:$90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan:$120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use$\,P\,$for the number of devices that need data plans as part of their cost. Find the model of the total cost of the Family Plan. Find the model of the total cost of the Mobile Share Plan. Assuming they stay under their data limit, find the number of devices that would make the two plans equal in cost. If a family has 3 smart phones, which plan should they choose? For exercises 17 and 18, use this scenario: A retired woman has$50,000 to invest but needs to make $6,000 a year from the interest to meet certain living expenses. One bond investment pays 15% annual interest. The rest of it she wants to put in a CD that pays 7%. If we let$\,x\,$be the amount the woman invests in the 15% bond, how much will she be able to invest in the CD? Set up and solve the equation for how much the woman should invest in each option to sustain a$6,000 annual return.

Two planes fly in opposite directions. One travels 450 mi/h and the other 550 mi/h. How long will it take before they are 4,000 mi apart?

Ben starts walking along a path at 4 mi/h. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at 6 mi/h. How long will it be before Amanda catches up to Ben?

Fiora starts riding her bike at 20 mi/h. After a while, she slows down to 12 mi/h, and maintains that speed for the rest of the trip. The whole trip of 70 mi takes her 4.5 h. For what distance did she travel at 20 mi/h?

A chemistry teacher needs to mix a 30% salt solution with a 70% salt solution to make 20 qt of a 40% salt solution. How many quarts of each solution should the teacher mix to get the desired result?

Paul has $20,000 to invest. His intent is to earn 11% interest on his investment. He can invest part of his money at 8% interest and part at 12% interest. How much does Paul need to invest in each option to make get a total 11% return on his$20,000?

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus$.10/mi driven. Plan B charges $100/wk plus$.05/mi driven.

Write the model equation for the cost of renting a truck with plan A.

Write the model equation for the cost of renting a truck with plan B.

Find the number of miles that would generate the same cost for both plans.

If Tim knows he has to travel 300 mi, which plan should he choose?

For the following exercises, use the given formulas to answer the questions.

$A=P\left(1+rt\right)\,$is used to find the principal amount Pdeposited, earning r% interest, for t years. Use this to find what principal amount P David invested at a 3% rate for 20 yr if$\,A=\text{\}8,000.$

The formula$\,F=\frac{m{v}^{2}}{R}\,$relates force (F), velocity (v), mass (m), and resistance (R). Find$\,R\,$when$\,m=45,$$v=7,$and$\,F=245.$

$F=ma\,$indicates that force (F) equals mass (m) times acceleration (a). Find the acceleration of a mass of 50 kg if a force of 12 N is exerted on it.

$Sum=\frac{1}{1-r}\,$is the formula for an infinite series sum. If the sum is 5, find$\,r.$

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

Solve for W:$\,P=2L+2W$

Use the formula from the previous question to find the width,$\,W,$of a rectangle whose length is 15 and whose perimeter is 58.

Solve for$\,f:\frac{1}{p}+\frac{1}{q}=\frac{1}{f}$

Use the formula from the previous question to find$\,f\,$when$\,p=8\,\text{and }q=13.$

Solve for$\,m\,$in the slope-intercept formula:$\,y=mx+b$

Use the formula from the previous question to find$\,m\,$when the coordinates of the point are$\,\left(4,7\right)\,$and$\,b=12.$

The area of a trapezoid is given by$\,A=\frac{1}{2}h\left({b}_{1}+{b}_{2}\right).\,$Use the formula to find the area of a trapezoid with$\,h=6,\text{ }{b}_{1}=14,\text{ and }{b}_{2}=8.$

Solve for h:$\,A=\frac{1}{2}h\left({b}_{1}+{b}_{2}\right)$

Use the formula from the previous question to find the height of a trapezoid with$\,A=150,\text{ }{b}_{1}=19,\text{ and }{b}_{2}=11.$

Find the dimensions of an American football field. The length is 200 ft more than the width, and the perimeter is 1,040 ft. Find the length and width. Use the perimeter formula$\,P=2L+2W.$

Distance equals rate times time,$\,d=rt.\,$Find the distance Tom travels if he is moving at a rate of 55 mi/h for 3.5 h.

Using the formula in the previous exercise, find the distance that Susan travels if she is moving at a rate of 60 mi/h for 6.75 h.

What is the total distance that two people travel in 3 h if one of them is riding a bike at 15 mi/h and the other is walking at 3 mi/h?

If the area model for a triangle is$\,A=\frac{1}{2}bh,$find the area of a triangle with a height of 16 in. and a base of 11 in.

Solve for h:$\,A=\frac{1}{2}bh$

Use the formula from the previous question to find the height to the nearest tenth of a triangle with a base of 15 and an area of 215.

The volume formula for a cylinder is$\,V=\pi {r}^{2}h.\,$Using the symbol$\,\pi \,$in your answer, find the volume of a cylinder with a radius,$\,r,$of 4 cm and a height of 14 cm.

Solve for h:$\,V=\pi {r}^{2}h$

Use the formula from the previous question to find the height of a cylinder with a radius of 8 and a volume of$\,16\pi$

Solve for r:$\,V=\pi {r}^{2}h$

Use the formula from the previous question to find the radius of a cylinder with a height of 36 and a volume of$\,324\pi .$

The formula for the circumference of a circle is$\,C=2\pi r.\,$Find the circumference of a circle with a diameter of 12 in. (diameter = 2r). Use the symbol$\,\pi \,$in your final answer.

Solve the formula from the previous question for$\,\pi .\,$Notice why$\,\pi \,$is sometimes defined as the ratio of the circumference to its diameter.

### Glossary

area
in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region:$\,A=LW$
perimeter
in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle:$\,P=2L+2W$
volume
in cubic units, the volume measurement includes length, width, and depth:$\,V=LWH$