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CHAPTER

Modeling a Business

Business, more than any other occupation, is a continual dealing with the future; it is a continual calculation, an instinctive exercise in foresight.

2-1 2-2 2-3 2-4 2-5 2-6 2-7 2-8

Interpret Scatterplots Linear Regression Supply and Demand Fixed and Variable Expenses Graphs of Expense and Revenue Functions Breakeven Analysis The Proﬁt Equation Mathematically Modeling a Business

Henry R. Luce, Publisher and Philanthropist

think What do you is quote? th in t n a e m Henry Luce What do you think? Answers should contain a reference to the fact that in business, it is important to look ahead, to predict, and to calculate and weigh all options when making decisions.

TEACHING RESOURCES Instructor’s Resource CD ExamView ® CD, Ch. 2 eHomework, Ch. 2 www.cengage.com/ school/math/ ﬁnancialalgebra

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If you do an Internet search for the word model, you will ﬁnd it used contexts. There are com computer models, medical models, in a variety of contexts working models, scientiﬁc models, business models, data models, and more. In each of these cases, a model is a representation. It is a well-formulated plan that can be used to represent a situation. Often, the model is a simpliﬁed version of the actual scenario. In this chapter, you will learn how to use mathematics to model a new business venture. You will use variables and examine relationships among those variables as you explore the business concepts of expense, revenue, and proﬁt. Statistics play an important role in the creation and use of a business model. You will represent real-world situations with equations, analyze the data that has been derived from those equations, use data to make predictions, and generate and interpret graphs in order to maximize proﬁt. As you work through this chapter, you will realize the importance of Henry Luce’s words. The models you explore will examine the future success or failure of a product and will help you make decisions that will mold and shape its future.

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Really?

Think of all the successful brand name products you use on a daily basis. Imagine all of the brain power and creative energy that went into inventing, perfecting, advertising, and selling these items. With all the resources major corporations have to make sure a product will sell, can you ever imagine a successful corporation producing a product that is a failure? The business world is full of products that have failed. Some of the failures are from very well-known companies, such as The Coca-Cola Company, Ford Motor Company, and Sony Corporation of America. Below are three famous failures. • New Coke In April 1985, The Coca-Cola Company changed the recipe of Coca-Cola, and the public was outraged. By July 1985, the company decided to bring back the original formula for Coca-Cola.

•

Edsel The Ford Motor Company unveiled its new 1958 Edsel in the fall of 1957. The car had a radically different look and was often described as ugly. The buying public did not like the sound of the name, and the advertising campaign had flaws. By 1959, production ceased on the Edsel, and the term is still used as a synonym for a failed design.

•

Sony Betamax In the late 1970s, Sony offered the first home video tape recorder, called the Betamax. Other manufacturers soon offered their own units. The units offered by other manufacturers all used the VHS tape format, and movies could be played on any brand’s machine, except Sony’s machine. Eventually, Sony had to abandon Betamax and issued its own VHS format tape recorders.

In this chapter, mathematical modeling is used to model a business. Point out that the word model can be used as a noun or a verb. You can create a mathematical model, or model a reallife situation mathematically. Students learn to graph and interpret scatterplots, linear functions, and parabolas.

REALLY? REALLY! Encourage students to look on the Internet to ﬁnd other product failures. Even Albert Einstein had failed inventions. Have students ﬁnd the failed inventions of Einstein. Make sure students understand the difference between a failed invention and a product that just doesn’t appeal to consumers.

Really!

© CAR CULTURE/CORBIS ©GUENTERMANAUS, 2009

You can do online research to find more details on these and other famous product failures.

CHAPTER OVERVIEW

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To guess is cheap. To guess wrongly is expensive. Ancient Chinese Proverb

2-1 Interpret Scatterplots Objectives

Key Terms

• Graph bivariate data.

• • • • •

• Interpret trends based on scatterplots.

data univariate data bivariate data scatterplot trend

• • •

correlation positive correlation negative correlation

• • •

causal relationship explanatory variable response variable

• Draw lines and curves of best ﬁt. EXAMINE THE QUESTION Show students how a scatterplot can depict a trend, and how trends affect business decisions. Give students several examples of univariate and bivariate data.

CLASS DISCUSSION Ask students to give a deﬁnition of function. Provide examples of scatterplots that are functions and scatterplots are not functions. Have students identify the differences.

How do scatterplots display trends? Any set of numbers is called a set of data. A single set of numbers is called univariate data. When a business owner keeps a list of monthly sales amounts, the data in the list is univariate data. Data that lists pairs of numbers and shows a relationship between the paired numbers is called bivariate data. If a business owner keeps records of the number of units sold each month and the monthly sales amount, the set is bivariate data. A scatterplot is a graph that shows bivariate data using points on a graph. Scatterplots may show a general pattern, or trend, within the data. A trend means a relationship exists between the two variables. A trend may show a correlation, a or association, between two variables. A positive correlation exists if the value of pos one variable increases when the value of the oth increases. A negative correlation other exi exists if the value of one variable decreases wh when the value of the other variable in increases. A trend may also show a causal relationship, which means ca on variable caused a change in the other one va variable. The variable which causes th the change in the other variable is the explanatory variable. The affected varie a able is the response variable. While a t trend may indicate a correlation or a causal r relationship, it does not have to. If two v variables are correlated, it does not mean that one caused the other. .COM UTTERSTOCK

ENSE FROM SH SED UNDER LIC APHY, 2009/U

GR © GEMPHOTO

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Skills and Strategies You can graph a scatterplot by hand. You can also graph a scatterplot on a graphing calculator.

EXAMPLE 1 Rachael runs a concession stand at the park, where she sells water bottles. She keeps a list of each day’s high temperature and the number of water bottles she sells each day. Rachael is looking for trends that relate the daily high temperature to the number of water bottles she sells each day. She thinks these two variables might be related and wants to investigate possible trends using a scatterplot. Below is the list of her ordered pairs. (65, 102), (71, 133), (79, 144), (80, 161), (86, 191), (86, 207), (91, 235), (95, 237), (100, 243) Construct a scatterplot by hand on graph paper. Then enter the data in a graphing calculator to create a scatterplot.

SOLUTION In each ordered pair, the ﬁrst number is the high temperature for the day in degrees Fahrenheit. The second number is the number of water bottles sold. Think of these as the x- and y-coordinates. The scatterplot is drawn by plotting the points with the given coordinates.

y

As students learn about scatterplots, be sure they realize that a scatterplot is a graph of a set of ordered pairs that may or may not show a relationship. They likely remember that to show an equation, ordered pairs are graphed on a coordinate plane and then connected. Point out that points graphed on a scatterplot are not connected. The position of the points are examined to determine the existence and strength of a correlation. Spend time discussing the difference between correlation and causation. Students need to understand that just because two sets of data have a correlation, there may or may not be a causation between the sets.

EXAMPLE 1

260 Water Bottle Sales

TEACH

180

Discuss with students the likelihood that this problem involves causation—the temperature is responsible for an increase in the water sales.

140

Compare this example to the following scenario:

220

100

x

70

80 90 100 Temperature (˚F)

Choose a scale for each axis that allows the scatterplot to ﬁt in the required space. To choose the scale, look at the greatest and least numbers that must be plotted for each variable. Label the axes accordingly. Then plot each point with a dot. Notice that you do not connect the dots in a scatterplot.

As the sales of ice cream increase, the number of lifeguard rescues increases. Students should recognize that ice cream does not cause the need for rescues, but both increase as the temperature increases. When it gets hotter, more people swim (making for an increased need for lifeguard rescues), and more people eat ice cream.

Use the statistics features on your graphing calculator to graph the scatterplot. Your display should look similar to the one shown.

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Interpret Scatterplots

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CHECK YOUR UNDERSTANDING

■ CHECK YOUR UNDERSTANDING If the temperature reaches 68 degrees Fahrenheit tomorrow, about how many water bottles do you predict will be sold? Explain.

Answer approximately 120 water bottles

EXAMPLE 2 © COPRID, 2009/USED UNDER LICENSE FROM SHUTTERSTOCK.COM

Rachael wants to interpret the trend shown in the scatterplot. What do you notice about the relationship between temperature and water bottle sales? Is there an explanatory variable and a response variable?

SOLUTION

As the temperature rises, the water bottle sales generally increase. So, there is a correlation between the data. Because the y-values increase when the x-values increase, the correlation is positive. Additionally, the rise in temperature caused the increase in the number of bottles sold. Therefore, the temperature is the explanatory variable and the number of bottles sold is the response variable.

EXAMPLE 2 Ask students what they think the scatterplot would look like if temperature was compared to sweater sales. They should realize that as temperatures increase, sales of sweaters decrease.

CHECK YOUR UNDERSTANDING Answer Because sales decrease when the temperature increases, the correlation is negative. This correlation is probably causal, so temperature is the explanatory variable and hot chocolate sales is the response variable.

■ CHECK YOUR UNDERSTANDING A local coffee shop sells hot chocolate. The manager keeps track of the temperature for the entire year and the hot chocolate sales. A scatterplot is graphed with temperature on the horizontal axis and hot chocolate sales on the vertical axis. Do you think the scatterplot shows a positive or negative correlation? Is there causation? Explain.

EXAMPLE 3 Determine if the following scatterplot depicts a positive correlation or a negative correlation.

EXAMPLE 3 Compare the trends in the scatterplots to what students remember about slopes of lines. Explain that the sign of the “slope” of the scatterplot is the same as the sign of the correlation. Ask students to think of two variables that are positively correlated, but one variable does not cause the other.

CHECK YOUR UNDERSTANDING Answer Taller people generally have larger feet, so a positive correlation is expected. There is causation.

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As the x-values increase, the y-values decrease. Therefore, this scatterplot shows a negative correlation between the two variables.

SOLUTION

■ CHECK YOUR UNDERSTANDING A local medical school is studying growth of students in grades 1–12. The height of each student in inches and the length of each student’s foot in centimeters is recorded, and a scatterplot is constructed. Do you think the scatterplot shows a positive correlation or a negative correlation? Is there causation?

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EXAMPLE 4 An elementary school principal compiled the following data about ten students at Compsett Elementary School. The ﬁrst number represents a student’s height in inches. The second number is the student’s reading level. Create a scatterplot of the data. Do you think a person’s height causes a higher reading level?

SOLUTION

Height (inches)

The scatterplot shows a positive correlation.

Reading Level

48

5.8

63

9.2

49

5.5

43

4.1

46

6.1

55

7.6

59

8.1

60

10.0

47

4.9

50

7.7

EXAMPLE 4 A person’s height does not cause a higher reading level. Most likely, both height and reading level for elementary school students increase with age. Keep in mind that if two variables are correlated, they are associated in some way. The student’s height does not cause the reading levels to be a certain value.

■ CHECK YOUR UNDERSTANDING

Point out to students that reading level and height both increase due to their dependence on age. In this case, age is called a lurking variable.

CHECK YOUR UNDERSTANDING Answer Answers will

Think of an example of data that might have a negative correlation but there is no causation.

vary. Sample answer: miles a car is driven and balance of a loan on the car.

EXAMPLE 5

EXAMPLE 5

In the next lesson students will learn about describing a correlation numerically, using a correlation coefﬁcient.

The scatterplot at the right shows the relationship between the number of text messages made by each of ten juniors while studying for Mr. Galati’s chemistry test last week and their scores on the test. Describe the trends you see in the data.

CHECK YOUR UNDERSTANDING Answer Based on the

SOLUTION

As the number of text messages increases, test grades do not increase, so there is no positive correlation. As the number of text messages increases, test grades do not decrease, so there is no negative correlation. There is no trend in the data, so there is no correlation.

formula C = πd, as the diameter increases, the circumference increases.

■ CHECK YOUR UNDERSTANDING Students in a biology class measure the circumference and diameter of every tree on the school property. The students create a table of ordered pairs and plan to draw a scatterplot. Should there be a positive correlation, a negative correlation, or no correlation?

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Interpret Scatterplots

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Applications

To guess is cheap. To guess wrongly is expensive. Ancient Chinese Proverb TEACH

1. Use what you learned in this lesson to explain how the quote can be interpreted by a business person. See margin.

Vocabulary Review Remind students that the terms explanatory and response are used when causation is implied. Have students offer original pairs of variables, tell how they are correlated, and determine if there seems to be causation.

2. A scatterplot shows the number of days that have passed and the number of days left in a month. The explanatory variable is the number of days that have passed. The response variable is the number of days left. Is there a positive or negative correlation? Explain. Negative correlation; as the number of days that have passed increases, the number of days left decreases.

3. Examine each scatterplot. Identify each as showing a positive correlation, a negative correlation, or no correlation.

ANSWERS 1. When making a business decision, it is wise to use available data and graphs to analyze trends, rather than guessing and subsequently investing money on a “hunch.” 4c. hours-explanatory; paycheck-response 5a.

a.

b. positive correlation

c.

negative correlation

d. no correlation

e.

f. negative correlation

68

Year

Per Capita Income in Dollars

2002

30,838

2003

31,530

2004

33,157

2005

34,690

2006

36,794

2007

38,615

2008

39,751

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negative correlation

no correlation

4. In a–d, each set of bivariate data has a causal relationship. Determine the explanatory and response variables for each set of data. a. height and weight of a student height-explanatory; weight-response b. grade on a math test and number of hours the student studied hours-explanatory; grade-response c. number of hours worked and paycheck amount See margin. d. number of gallons of gas consumed and weight of a car weight-explanatory; gallons-response

5. The table shows the personal income per capita (per person) in the United States for seven selected years. a. Draw a scatterplot for the data. See margin. b. Describe the correlation. positive correlation

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6. The following set of ordered pairs gives the results of a science experiment. Twelve people were given different daily doses of vitamin C, in milligrams, for a year. This is the x-value. They reported the number of colds they got during the year. This is the y-value. (100, 4), (100, 4), (100, 3), (250, 3), (250, 2), (250, 2), (500, 1), (500, 2), (500, 1), (1,000, 1), (1,000, 2), (1,000, 1) a. Construct a scatterplot. See margin. b. Describe the correlation. negative correlation c. Should the scientists label the vitamin C intake the explanatory variable and the number of colds the response variable? Explain. Yes; it seems to be causal.

7. The enrollment at North Shore High School is Year Enrollment given in the table. In each year, the number of 2006 801 students on the baseball team was 19. 2007 834 a. If x represents the year and y represents the 2008 844 enrollment, draw a scatterplot to depict the 2009 897 data. See margin. b. Describe the correlation from the 2010 922 scatterplot. positive correlation c. If x represents the enrollment and y represents the number of students on the baseball team, draw a scatterplot to depict the data. See margin. d. Describe the correlation from the scatterplot. no correlation

TEACH Data Entry Students need to double check calculator list entries before making calculations. The calculator will display incorrect answers just as quickly as it will correct answers if data are entered incorrectly. It is very easy to make a mistake entering data. Also remind them to check the number of pieces of data they entered, to make sure it agrees with the original list.

ANSWERS 6a.

7a.

8. The MyTunes Song Service sells music downloads. Over the past few years, the service has lowered its prices. The table shows the price per song and the number of songs downloaded per day at that price. Price per Song

Number of Downloads (in thousands)

$0.89

1,212

$0.79

1,704

$0.69

1,760

$0.59

1,877

$0.49

1,944

$0.39

2,011

7c.

8a. As x decreases, y increases. 8b. There is a negative correlation.

a. Examine the data without drawing a scatterplot. Describe any trends you see. See margin. b. Draw a scatterplot. Describe the correlation. See margin. c. Approximate the number of downloads at a price of $0.54 per song. Explain your reasoning. See margin. 9. Perform an online search to answer the questions below. Answers vary. a. Find your state’s population for each of the last ten years. b. Create a table of bivariate data. Let x represent the year, and let y represent the population. c. Create a scatterplot for the data. d. Describe the correlation between the year and your state’s population.

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8c. Approximately 1,910,500 songs. The average of $0.59 and $0.49 is $0.54 and the average of 1,877 and 1,944 is 1,910.

Interpret Scatterplots

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The only useful function of a statistician is to make predictions, and thus provide a basis for action. William Edwards Deming, Professor and Statistician

2-2 Linear Regression Objectives

Key Terms

• Be able to ﬁt a regression line to a scatterplot.

• • • • •

• Find and interpret correlation coefﬁcients. • Make predictions based on lines of best ﬁt.

The trends shown by scatterplots can be used to predict the future. But making a prediction without a line of best ﬁt to guide you would be arbitrary. Point out that changes could make the prediction inaccurate. However, the predictions are based on current data and are better than haphazard guessing.

Put a scatterplot on the board and point to different x-values in the domain. For each, ask students if it is an example of interpolation or extrapolation. What can you tell about the sign of the correlation coefﬁcient and the slope of the regression line?

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Water Bottle Sales

CLASS DISCUSSION

Chapter 2

• • • • • •

interpolation extrapolation correlation coefﬁcient strong correlation weak correlation moderate correlation

How can the past predict the future?

EXAMINE THE QUESTION

70

line of best ﬁt linear regression line least squares line domain range

Many scatterplot points can be approximated by a single line that best ﬁts the scattered points. This line may be called a: line of best ﬁt, linear regression line, or least squares line. This line can be used to display a trend and predict corresponding variables for different situations. It is more efﬁcient to rely on the single line rather than the scatterplot points because the line can be represented by an equation. Recall that the domain is a set of ﬁrst elements of the ordered pairs, and the range is the set of corresponding second elements. Interpolation means to predict corresponding y-values, given an x-value within the domain. Extrapolation means to predict corresponding y-values outside of the domain. The scatterplot shown is from Example 1 in the previous lesson. The line shown is a line of best ﬁt because it closely follows the trend of the data points. The blue labels are included to identify the axes, but will not be shown on a calculator display. Generally, the distance the points lie from the line of best ﬁt determines how good a predictor the line is. If most of the points lie close to the line, the line is a better predictor of the trend of the data than if the points lie far from the line. If the points lie far from the line, the line is not good for predicting a trend. The correlation coefﬁcient, r, is a number between –1 and 1 incluTemperature (˚F) sive that is used to judge how closely the line ﬁts the data. Negative correlation coefﬁcients show negative correlations, and positive correlation coefﬁcients show positive correlations. If the correlation coefﬁcient is near 0, there is little or no correlation. Correlation coefﬁcients with an absolute value greater than 0.75 are strong correlations. Correlation coefﬁcients with an absolute value less than 0.3 are weak correlations. Any other correlation is a moderate correlation.

Modeling a Business

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Skills and Strategies The line of best ﬁt and the correlation coefﬁcient can be found using a graphing calculator.

EXAMPLE 1 Find the equation of the linear regression line for Rachael’s scatterplot in Example 1 from Lesson 2-1. Round the slope and y-intercept to the nearest hundredth. The points are given below.

TEACH To enter regression lines and have them graphed on the calculator’s display along with the scatterplot, students need to be sure they have the scatterplot and the equation set to display on the same screen.

(65, 102), (71, 133), (79, 144), (80, 161), (86, 191),

EXAMPLE 1

(86, 207), (91, 235), (95, 237), (100, 243)

Different calculators have keys in different locations with different names, but the basic functions are similar. Remind students to input data carefully. An incorrect entry results in an incorrect answer.

SOLUTION Although it is possible to ﬁnd the linear regression equation using paper and pencil, it is a lengthy process. Using the linear regression feature on a graphing calculator produces more accurate results. Enter the ordered pairs into your calculator. Then use the statistics menu to calculate the linear regression equation. The equation is of the form y = mx + b, where m is the slope and b is the y-intercept. Rounding the slope and y-intercept to the nearest hundredth, the equation of the regression line is y = 4.44x – 187.67. Note that calculators may use different letters to represent the slope or the y-intercept. Remember that the coefﬁcient of x is the slope.

■ CHECK YOUR UNDERSTANDING Find the equation of the linear regression line of the scatterplot deﬁned by these points: (1, 56), (2, 45), (4, 20), (3, 30), and (5, 9). Round the slope and y-intercept to the nearest hundredth.

CHECK YOUR UNDERSTANDING Answer y = –11.9x + 67.7

EXAMPLE 2 Review the units of the slopes for the examples shown throughout Lesson 2-1, so students understand that the slope is a rate of change.

CHECK YOUR UNDERSTANDING Answer 9 bottles

EXAMPLE 2 Interpret the slope as a rate for Rachael’s linear regression line. Use the equation from Example 1. ∆y SOLUTION The formula for slope is m = ___. The range val∆x ues, y, represent bottles sold and the domain values, x, repre-

© ROMAN SIGAEV, 2009/USED UNDER LICENSE FROM SHUTTERSTOCK.COM

sent temperatures. The slope is a rate of bottles per degree. The slope is 4.44, which means that for each one-degree increase in temperature, 4.44 more water bottles will be sold. Rachael cannot sell a fraction of a water bottle, so she will sell approximately 4 more bottles for each degree the temperature rises.

■ CHECK YOUR UNDERSTANDING Approximately how many more water bottles will Rachael sell if the temperature increases 2 degrees?

2-2

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Linear Regression

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EXAMPLE 3 Ask students if this is an example of interpolation or extrapolation.

EXAMPLE 3 Rachael is stocking her concession stand for a day in which the temperature is expected to reach 106 degrees Fahrenheit. How many water bottles should she pack?

SOLUTION SOL

The linear regression equation tells Rach Rachel the approximate number of bottles she sho should sell given a speciﬁc temperature. Substitute 106 for x in the equation, and compute y, the num number of water bottles she should expect to sell.

D UNDER VAL 20 09/USE © VASILIY KO CK.COM TO RS TE UT FROM SH

LICENSE

CHECK YOUR UNDERSTANDING

Eq Equation of the regression line reg

y = 4.44x – 187.67

Substitute 106 for x. Su

y = 4.44(106) – 187.67

Simplify. Si

y = 282.97

If the trend continues and the temperature re reaches 106 degrees Fahrenheit, Rachael should e expect to sell approximately 283 water bottles. S should stock 283 bottles. This is an example She becaus 106 degrees Fahrenheit was not between the of extrapolation because hi h and d low l l high x-values of the original domain.

■ CHECK YOUR UNDERSTANDING

Answer 181 water bottles; it is an example of interpolation.

How many water bottles should Rachael pack if the temperature forecasted were 83 degrees? Is this an example of interpolation or extrapolation? Round to the nearest integer.

EXAMPLE 4 Have students identify all correlations as strong, moderate, or weak. It is a common error for students to ignore negative signs when reading correlation coefﬁcients on calculator displays. Caution them to note the sign of the coefﬁcient.

CHECK YOUR UNDERSTANDING Answer r = –0.998; this coefﬁcient represents a strong negative correlation.

EXTEND YOUR UNDERSTANDING

EXAMPLE 4 Find the correlation coefﬁcient to the nearest hundredth for the linear regression for Rachael’s data. Interpret the correlation coefﬁcient.

SOLUTION Use a graphing calculator to ﬁnd the correlation coefﬁcient. Round r to the nearest hundredth.

r = 0.97

Because 0.97 is positive and greater than 0.75, there is a strong positive correlation between the high temperature and the number of water bottles sold.

■ CHECK YOUR UNDERSTANDING Find the correlation coefﬁcient to the thousandth for the linear regression for the data in Check Your Understanding for Example 1. Interpret the correlation coefﬁcient.

Answer The correlation is weak and negative.

■ EXTEND YOUR UNDERSTANDING Carlos entered data into his calculator and found a correlation coefﬁcient of – 0.28. Interpret this correlation coefﬁcient.

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Applications The only useful function of a statistician is to make predictions, and thus provide a basis for action. William Edwards Deming, Professor and Statistician TEACH

1. Apply what you have learned in this lesson to give an interpretation of the quote. See margin.

Investigate Correlation Coefﬁcient

2. Over the past four years, Reggie noticed that as the price of a slice of pizza increased, her college tuition also increased. She found the correlation coefﬁcient was r = 0.49. Which of the following scatterplots most accurately displays Reggie’s data? Explain. See margin.

At some point, have students switch the x and y variables and ﬁnd the correlation coefﬁcient. They should see it remains the same. However, the slope of the regression line changes.

ANSWERS a.

b.

3. In Exercise 2, would the price of a slice of pizza be labeled as the explanatory variable and the tuition as the response variable? Explain. No, there is no apparent causation. Both prices may have increased due to inﬂation over time.

4. The table gives enrollments at North Shore High School. a. Find the equation of the regression line. Round the slope and y-intercept to the nearest hundredth. y = 30.5x – 60,384.4 b. What is the slope of the linear regression line? 30.5 c. What are the units of the slope expressed as a rate? students per year d. Based on the linear regression line, how many students will be enrolled in the year 2016? Round to the nearest integer. 1,104 students e. Is your answer to part d an example of interpolation or extrapolation? Explain. Extrapolation; 2016 is outside of the domain. f. Find the correlation coefﬁcient to the nearest hundredth. r = 0.98 g. Describe the correlation. strong positive correlation 5. Examine the data from Exercise 4. a. Find the mean (arithmetic average) of the ﬁve years. 2008 b. Find the mean of the ﬁve enrollment ﬁgures. 859.6 c. Create an ordered pair whose x-value is the mean of the years and whose y-value is the mean of the enrollments. (2008, 859.6) d. Show that the ordered pair satisﬁes the linear regression equation. What does this mean regarding the regression line? See margin. 6. Describe each of the following correlation coefﬁcients using the terms strong, moderate, or weak and positive or negative. a. r = 0.21 weak positive b. r = – 0.87 strong negative c. r = 0.55 moderate positive d. r = – 0.099 weak negative e. r = 0.99 strong positive f. r = – 0.49 moderate negative

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1. The ability to make predictions is valuable in many ﬁelds. It allows you to be proactive in regards to the future, rather than just assuming you have no control over the future. 2. Graph a shows a moderate correlation that could

Year

Enrollment

2006

801

2007

834

2008

844

2009

897

2010

922

be described by r = 0.49. Graph b is almost linear and shows a strong correlation. 5d. y = 30.5x – 60,384.4 y = 30.5(2008) – 60,384.4 = 859.6. The linear regression line always passes through the point formed by the averages of each set of data.

Linear Regression

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TEACH Examine Scatterplots For all examples, ask students about the shape of the scatterplots (linear or not), the sign of the association (positive or negative), and the strength (strong, moderate, weak) of the correlation. They should get comfortable using these terms to describe scatterplots.

Amount of Restaurant Bill ($)

Tip Amount ($)

45.55

7.00

52.00

15.00

66.00

6.00

24.44

6.00

57.90

15.00

89.75

23.00

33.00

8.00

ANSWERS 8c. tip dollars per restaurant bill amount 9. A positive slope means the line is increasing, which means as x increases, y increases. If y increases as x increases, the correlation coefﬁcient is positive. A negative slope means the line is decreasing, which means as x increases, y decreases. If y decreases as x increases, the correlation coefﬁcient is negative. 11. Yes, if the points are linear, the regression line will go through every point. If the points are not linear, it may not go through any point on the scatterplot.

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7. The table gives the number of songs downloaded from MyTunes at different prices per song. a. Find the equation of the linear regression line. Round the slope and y-intercept to the nearest hundredth.

Price per Song

Number of Downloads (in thousands)

$0.89

1,212

$0.79

1,704

y = –1,380.57x + 2,634.90 $0.69 1,760 b. What is the slope of the linear $0.59 1,877 regression line? –1,380.57 $0.49 1,944 c. What are the units of the slope when $0.39 2,011 it is expressed as a rate? thousands of downloads per dollar d. Based on the linear regression line, how many thousands of downloads would MyTunes expect if the price was changed to $0.45? Round to the nearest integer. 2,014 e. Is your answer to part d an example of interpolation or extrapolation? interpolation f. Find the correlation coefﬁcient to the nearest hundredth. r = – 0.90 g. Describe the correlation. strong negative correlation

8. Julie is a waitress. On the left is a log of her tips for yesterday’s shift. a. Find the equation of the linear regression line. Round the slope and y-intercept to the nearest hundredth. y = 0.22x – 0.27 b. What is the slope of the linear regression line? 0.22 c. What are the units of the slope when it is expressed as a rate? See margin. d. Based on the linear regression line, what tip would Julie receive if the restaurant bill were $120? Round to the nearest dollar. $26 e. Is your answer to part d an example of interpolation or extrapolation? Explain. Extrapolation; $120 is outside of the original domain. f. Find the correlation coefﬁcient for this data. Round to the nearest hundredth. r = 0.75 g. Describe the correlation. moderate positive correlation h. Based on the linear regresA B sion line, Julie creates a 1 Restaurant Bill Predicted Tip spreadsheet to compute 2 predicted tips for any restaurant bill amount. Write the 3 formula that can be used to 4 compute the predicted tips 5 in column B. =A2*0.22–0.27 9. Explain why the sign of the slope of a regression line must be the same as the sign of the correlation coefﬁcient. See margin. 10. Which of the following scatterplots shows a correct line of best ﬁt? c

a.

b.

c.

11. Is it possible for a linear regression line to go through every point on the scatterplot? Is it possible for a linear regression line to not go through any point on the scatterplot? See margin.

Modeling a Business

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Teach a parrot the terms “supply and demand” and you’ve got an economist. Thomas Carlyle, Philosopher

Supply and Demand 2-3 Key Terms • • •

widget function demand function

Objectives • • •

demand supply wholesale price

• • • •

markup retail price equilibrium shift

• Understand the slopes of the supply and demand curves. • Find points of equilibrium.

How do manufacturers decide the quantity of a product they will produce?

EXAMINE THE QUESTION

Economists often call a new, unnamed product a widget. If a business develops a new product, the number of items they need to manufacture is a key question they need to address. Graphs may be used to help answer this question. Such graphs compare the price of an item, p, and the quantity sold, q. The horizontal axis is labeled p and the vertical axis is labeled q. The set of p values on the horizontal axis is the domain. The set of q values on the vertical axis is the range. A function is a rule that assigns a unique member of the range to each element of the domain. Graphs using the p- and q- axes can be q used to model a trend in consumers’ inter- q est in a product. A demand function relates the quantity of a product to its price. If a widget has a low price, many people may want it and will be able to afford it, so a large quantity may be sold. p If it has a high price, fewer widgets will be sold. As the price increases, demand (the quantity consumers want) is likely to decrease, and as price decreases, demand increases. The graph of the demand function has a negative slope. However, its curvature varies. Graphs using p- and q- axes can also be used to model a trend in the manufacturing of a product. Producers provide supply (the quantity of items available to be sold). If a widget q q sells for a high price, the manufacturer may be willing to produce many items to maximize proﬁt. If the widget sells for a lower price, the manufacturer may produce less. As price increases, supply p increases. The graph of the supply function has a positive slope. Its curvature also varies.

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Have students imagine they are starting a new business. What are the dangers of overproducing? (excess inventory and wasted money) What are the dangers of underproducing? (lost income) Deciding how many items to produce is not guesswork.

q

p

p

q

p

Supply and Demand

p

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CLASS DISCUSSION Give students other examples of functions, including non-numerical examples. For example, the initials function assigns ﬁrst initials to people. The hair color function assigns a color to a person based on their hair color. Use examples such as these to show that two different elements of the domain could be mapped to the same element of the range. Ask students about any products that are difﬁcult to ﬁnd because stores don’t always have them in stock.

TEACH Supply and demand are concepts that impact students’ lives, yet they have probably never given much thought to them. Be sure they can give examples of the supply and demand for products they typically purchase or use.

EXAMPLE 1 Point out to students that the markup is not all proﬁt. Markup costs are used to cover the store’s expenses. Ask them to identify some of these expenses, which include rental costs, paying employees, advertising, utility costs, and purchasing store ﬁxtures.

CHECK YOUR UNDERSTANDING Answer r – x Sometimes students have trouble expressing literal subtraction expressions, and they misplace the minuend and the subtrahend.

Economic decisions require research, and knowledge of the laws of supply and demand. To examine the law of supply and demand, graph both functions on the same axes. Examine what happens as the manufacturer sets different prices. Keep in mind that the manufacturer sells the items to retailers, such as stores, and not directly to the general public. The price the manufacturer charges the retailer is the wholesale price. Retailers increase the price a certain amount, called markup, so the retailer can make a proﬁt. The price the retailer sells the item to the public for is the retail price.

q

Supply

h

q

Supply

h k

k p

Supply

h

Demand

3

q

Demand

7

p

Demand

6

p

Look at the graph on the left. If the price is set at $3.00, consumers will demand h widgets, and manufacturers will be willing to supply k widgets. There will be a shortage of widgets. High demand and low supply often create a rise in price. The manufacturer knows that people want widgets, yet there are not enough of them. If the price rises, demand will fall. Look at the center graph. If the price is set at $7.00, consumers will demand k widgets, and manufacturers will be willing to supply h widgets. There will be too much supply, and manufacturers will have to lower the price to try and sell the high inventory of widgets. Look at the graph on the right. At a price of $6.00, the number of widgets demanded by consumers is equal to the number of widgets manufacturers will supply. Where the functions of supply and demand intersect, the market is in equilibrium.

Skills and Strategies Supply depends on the expenses involved in producing a widget and the price for which it can be sold. The factors of supply will be outlined in Lesson 2-4. This lesson concentrates on demand.

EXAMPLE 1 The Wacky Widget Company sells widgets for $2.00 each wholesale. A local store has a markup of $1.59. What is the retail price?

SOLUTION

Add the markup to the wholesale price.

Markup + Wholesale price = Retail price

2.00 + 1.59 = 3.59

The retail price is $3.59.

■ CHECK YOUR UNDERSTANDING The wholesale price of an item is x dollars. The retail price is r dollars. Express the markup algebraically.

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EXAMPLE 2

EXAMPLE 2

Point out to students that a 100% markup is the same as doubling the wholesale price.

The Robear Corporation sells teddy bears at a wholesale price of $23.00. If a store marks this up 110%, what is the retail price?

SOLUTION Compute the markup amount. Then calculate the retail price. Markup rate × Wholesale price = Markup

1.10 × $23 = $25.30

Wholesale price + Markup = Retail price

$23.00 + $25.30 = $48.30

CHECK YOUR UNDERSTANDING Answer x + 0.9x, or 1.09x.

EXAMPLE 3

The retail price is $48.30.

Point out the importance of labeling the axes and advise students to label all problems.

■ CHECK YOUR UNDERSTANDING A banner company sells 5-foot banners to retailers for x dollars. The St. James Sign Shop marks them up 90%. Express the retail price at the St. James store algebraically.

CHECK YOUR UNDERSTANDING Answer Supply will exceed demand. Suppliers will attempt to sell the excess widgets by lowering the price.

EXAMPLE 3 The graph shows the supply and demand curves for a widget. Explain what happens if the price is set at $9.00.

q Supply

EXAMPLE 4

SOLUTION Because $9.00 is less than the equilibrium price, demand will exceed supply. Suppliers will attempt to sell the widget at a higher price.

Demand p

11

■ CHECK YOUR UNDERSTANDING Use the graph to explain what happens if the price is set at $15.00.

EXAMPLE 4 A company wants to base the price of its product on demand for the product, as well as on expenses. It takes a poll of several of its current retailers to ﬁnd out how many widgets they would buy at different wholesale prices. The results are shown in the table. The company wants to use linear regression to create a demand function. What is the equation of the demand function? Round the slope and y-intercept to the nearest hundredth.

SOLUTION

Use the linear regression feature on your graphing calculator. The equation is q = –1,756.19p + 30,238.82. This represents the demand function.

■ CHECK YOUR UNDERSTANDING Explain why it makes sense that the demand function has a negative slope.

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Remind students that the linear regression line is an “average” line, and if they substitute the wholesale prices from the table into the regression equation, they will not get the quantity shown in the table, unless the regression line goes through the point with those coordinates. That is unlikely.

Wholesale Price ($) (p)

Quantity Retailers Would Purchase (in thousands) (q)

15.25

3,456

15.50

3,005

15.75

2,546

16.00

2,188

16.25

1,678

16.50

1,290

16.75

889

17.00

310

CHECK YOUR UNDERSTANDING Answer As the price increases, less quantity is demanded, so the demand function slopes downward. This is indicated by a negative slope.

Supply and Demand

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Applications Teach a parrot the terms “supply and demand” and you’ve got an economist. Thomas Carlyle, Philosopher TEACH

1. Interpret the quote in the context of what you learned in this lesson.

Shifts of Demand Curves Notice that when a demand curve is shifted to the right, some students may see this as shifted “up.” Similarly, when a curve is shifted left, it may appear to be shifted “down.” Try to have students use left and right to describe the shifts.

ANSWERS 1. Parrots repeat the things they hear over and over. Supply and demand is such an important concept in economics that it is constantly used by economists. 5b. Demand will exceed supply, and suppliers will attempt to sell the product at a higher price. 5e. Supply will exceed demand, and suppliers will attempt to sell the excess product at a lower price. 6b. Demand will exceed supply, and suppliers will attempt to sell the product at a higher price. 6c. Supply will exceed demand, and suppliers will attempt to sell the excess product at a lower price.

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See margin.

2. An automobile GPS system is sold to stores at a wholesale price of $97. A popular store sells them for $179.99. What is the store’s markup? $82.99 3. A CD storage rack is sold to stores at a wholesale price of $18. a. If a store has a $13 markup, what is the retail price of the CD rack? $31 b. Find the percent increase of the markup to the nearest percent. 72% 4. A bicycle sells for a retail price of b dollars from an online store. The wholesale price of the bicycle is w. a. Express the markup algebraically. b – w b–w b. Express the percent increase of the markup algebraically. ______ • 100 w 5. The graph shows supply and demand curves for the newest SuperWidget. q

Supply

y

x $0.98 $1.12

a. b. c. d. e.

Demand p

What is the equilibrium price? $1.12 What will happen if the price is set at $0.98? See margin. How many SuperWidgets are demanded at a price of $0.98? y How many SuperWidgets are supplied at a price of $0.98? x What will happen if the price is set at $1.22? See margin.

6. The graph below shows supply and demand curves for a new mp3 player accessory. a. What is the equilibrium price? $35 b. Describe the relationship of supply and demand if the item were sold for $20. See margin. q c. Describe the relationship of supply Supply and demand if the item were sold for $40. See margin. d. Name the domain that will increase demand. x < 35 Demand e. Name the domain that will increase p 35 supply. x > 35

Modeling a Business

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7. The supply and demand curves for a new widget are shown in the graph. Notice there are two demand curves. The original demand curve is d1. Months after the product was introduced, there was a possible health concern over use of the product, and demand dropped to the new demand curve, d2. The movement of the demand curve is called a shift. a. What was the equilibrium price before the demand shift? b b. What was the equilibrium quantity before the demand shift? e c. What was the equilibrium price after the demand shift? a d. What was the equilibrium quantity q Supply after the demand shift? c e. Express algebraically the difference in quantity demanded at price b before and after the shift. e – c e f. Copy a rough sketch of the graph c into your notebook. Label the curves. d d2 1 Where would the demand curve have p a b shifted if a health beneﬁt of the new widget was reported? It would shift to the right. 8. Debbie is president of a company that produces garbage cans. The company has developed a new type of garbage can that is animalproof, and Debbie wants to use the demand function to help set a price. She surveys ten retailers to get an approximation of how many garbage cans would be demanded at each price, and creates a table. a. Find the equation of the linear regression line. Round the slope and y-intercept to the nearest hundredth. q = –136.08p + 2,535.79 b. Give the slope of the regression line and interpret the slope as a rate. See margin. c. Find the correlation coefﬁcient and interpret it. Round to the nearest hundredth. r = –0.99; there is a strong negative correlation. d. Based on the linear regression line, how many garbage cans would be demanded at a wholesale price of $18.00? Round to the nearest hundred garbage cans. 86 hundred e. Was your answer to part d an example of extrapolation or interpolation? Explain. Extrapolation, $18.00 is not in the domain. f. Look at your answer to part d. If the company sold that many garbage cans at $18.00, how much money would the company receive from the garbage can sales? $154,800

TEACH Multi-part Exercises These problems have many parts that depend on previous parts being answered correctly. Stress the fact that calculator list entries must be checked carefully.

ANSWERS 8b. The slope is –136.08. As a rate, the slope is expressed as garbage cans per dollar. For each dollar increase in price, about 136 less garbage cans are demanded.

Wholesale Price ($) (p)

Quantity Demanded By Retailers (in hundreds) (q)

13.00

744

13.50

690

14.00

630

14.50

554

15.00

511

15.50

456

16.00

400

16.50

300

17.00

207

17.50

113

9. A company that produces widgets has found its demand function to be q = –1,500p + 90,000. a. For each dollar increase in the wholesale price, how many fewer widgets are demanded? 1,500 b. How many widgets would be demanded at a price of $20? 60,000 c. How many widgets would be demanded at a price of $21? 58,500 d. What is the difference in quantity demanded caused by the $1 increase in wholesale price? 1,500 less e. The company sets a price of $22.50. How many widgets will be demanded? 56,250 f. How much will all of the widgets cost the store to purchase at a price of $22.50? $1,265,625 g. If the store marks up the widgets that cost $22.50 at a rate of 50%, what is the retail price of each widget? $33.75

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An economist is an expert who will know tomorrow why the things he predicted yesterday didn’t happen today. Laurence J. Peter, Professor

2-4 Fixed and Variable Expenses Objectives

Key Terms

• Understand the difference between ﬁxed and variable expenses.

• • • •

• Create an expense equation based on ﬁxed and variable expenses.

Students have possibly never brainstormed on all of the variables that affect the manufacturing process. Give them time to think and formulate answers before they read the text.

Can you think of any other expenses involved in the manufacturing process? Point out that revenue depends on quantity sold, and the quantity sold depends on the price charged, so revenue ultimately depends on the price charged.

• • • •

revenue equation proﬁt loss breakeven point

What expenses are involved in the manufacturing process?

EXAMINE THE QUESTION

CLASS DISCUSSION

variable expenses ﬁxed expenses expense equation revenue

A group of art school students have decided to start a business producing hand-painted jeans. They made a list of expenses for running the business. Some of these items must be purchased while others may be rented. • factory space • telephone • paint • furniture • computer • packaging • delivery trucks • ofﬁce supplies • postage • electricity • jeans • labor All of these expenses fall into one of two categories—variable expenses or ﬁxed expenses. Expenses that depend on the number of items produced are variable expenses. Examples of variable expenses are costs for raw materials such as jeans, paint, ofﬁce supplies, and labor expenses, because these costs change based on the quantity produced. Some expenses do not change based on the quantity produced. These expenses are ﬁxed expenses. Examples of ﬁxed expenses are the cost of the furniture and the computer. The cost of having the lights on is the same regardless of how many items are produced, so this is also a ﬁxed expense. The total expenses is the sum of the ﬁxed and variable expenses. The expense equation is E = V + F where E represents total expenses, V represents variable expenses, and F represents ﬁxed expenses The income a business receives from selling its product is revenue. Revenue is the price for which each product was sold times the number of products sold. The revenue equation is R = pq where R represents revenue, p represents the price of the product, and q represents the quantity of products sold The difference obtained when expenses are subtracted from revenue is a proﬁt when positive and a loss when negative. When the expenses and the revenue are equal, there is no proﬁt or loss. This is the breakeven point.

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Skills and Strategies Business owners use mathematical models to analyze expenses and revenue to determine proﬁtability of a product.

EXAMPLE 1 The art students have researched all of their potential expenses. The ﬁxed expenses are $17,600. The labor and materials required for each pair of painted jeans produced cost $7.53. Represent the total expenses as a function of the quantity produced, q. If q units are produced at a cost of $7.53 per unit, the variable expenses can be represented by:

SOLUTION

V = 7.53q The ﬁxed expenses, $17,600, do not depend on the quantity produced. The total expenses, E, are the sum of the variable and the ﬁxed expenses. E=V+F E = 7.53q + 17,600

TEACH Relate what students know about variables and constants to the new vocabulary terms of variable expenses and ﬁxed expenses.

EXAMPLE 1 Point out to students that the variable expenses part of the expense equation needs a variable. The ﬁxed part of the equation is also called a constant.

CHECK YOUR UNDERSTANDING Answer $6.00

The total expenses, E, are a function of the quantity produced, q.

■ CHECK YOUR UNDERSTANDING A widget manufacturer’s expense function is E = 6.00q + 11,000. What are the variable costs to produce one widget?

EXAMPLE 2 Kivetsky Ski Supply manufactures hand warmers for skiers. The expense function is E = 1.18q + 12,000. Find the average cost of producing one pair of hand warmers if 50,000 hand warmers are produced.

SOLUTION

Find the total cost of producing 50,000 hand warmers by substituting 50,000 for q in the expense function. E = 1.18q + 12,000

Ask students the costs of producing 500 widgets and 501 widgets. Find the difference and compare it to the slope of the expense function.

EXAMPLE 2 Point out to students that intuitively, the average cost decreases as the number of items produced increases, because the ﬁxed costs are “spread out” over many items.

CHECK YOUR UNDERSTANDING Answer

Substitute.

E = 1.18(50,000) + 12,000

Multiply.

E = 59,000 + 12,000

Simplify.

E = 71,000

3.40q + 189,000 _______________ q

To ﬁnd the average cost, divide the total cost by the number produced. Divide by 50,000.

71,000 _______ = 1.42 50,000

The average cost to produce one hand warmer is $1.42 when 50,000 are produced.

■ CHECK YOUR UNDERSTANDING The expense function for a certain product is E = 3.40q + 189,000. Express the average cost of producing q items algebraically.

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EXAMPLE 3 Remind students that revenue depends on quantity sold, and the quantity sold depends on the price charged, so revenue depends on price. As a result, they can express revenue in terms of the variable p.

EXAMPLE 3 Willie’s Widgets has created a demand function for its widgets, where q is the quantity demanded and p is the price of one widget. q = –112p + 4,500 Its expense function is E = 3.00q + 18,000. Express the expense function as a function in terms of p. Because E is a function of q, and q is a function of p, express E as a function of p using substitution.

SOLUTION

CHECK YOUR UNDERSTANDING

E = 3.00q + 18,000

Answer E = –16.00p + 90,000 You can give students similar problems to make sure they are substituting correctly.

Substitute for q.

E = 3.00(–112p + 4,500) + 17,600

Distribute.

E = –336p + 13,500 + 17,600

Simplify.

E = –336p + 31,100

The expense function in terms of the price, p, is E = –336p + 31,100.

Remind students to label the axes and to use these labels. Since the y-coordinates (the “heights”) represent dollars, the revenue and expense heights can be compared to see which is greater. You may put the graph on the board, and pick a speciﬁc value of q. Highlight the revenue and expenses at this q-value, and draw a vertical line connecting these two points. Explain that the length of this line represents the difference between revenue and expenses, and is a graphical representation of proﬁt.

■ CHECK YOUR UNDERSTANDING A corporation’s expense function is E = 4.00q + 78,000. The demand function was determined to be q = –4p + 3,000. Express E in terms of p.

EXAMPLE 4 Wally’s Widget World created a monthly expense equation, E = 1.10q + 4,200. Wally’s Widget World plans to sell its widgets to retailers at a wholesale price of $2.50 each. How many widgets must be sold to reach the breakeven point?

SOLUTION

Use a graphing calculator. Graph the revenue function, R = 2.50q, using R as revenue and q as the quantity sold. Notice that q is the independent variable, so the horizontal axis is labeled q. The labels on the graph below will not be shown on the calculator display. Graph the revenue function on a coordinate grid.

dollars

EXAMPLE 4

R = 2.50q

Graph the expense function on the same coordinate grid. Use your calculator’s graph intersection feature to ﬁnd the point of intersection.

dollars

q

Intersection X=3000

Y=7500 q

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dollars

It will cost $7,500 to manufacture 3,000 widgets. If 3,000 widgets are sold to stores for $2.50 each, the company will receive $7,500. Notice that no proﬁt is made—the revenue equals the expense. The point (3,000, 7,500) is the breakeven point. Look at sections of the graph before and after the breakeven point to see if you understand what happens. Revenue

Answer Wally’s Widget World is operating below the breakeven point; expenses cost $140 more than what is made in revenue. For more practice, try other examples using other quantities.

Expense

7,500

CHECK YOUR UNDERSTANDING

EXAMPLE 5

q

3,000

If more than 3,000 widgets are produced, the company will be operating in the green area of the graph, where R > E. A proﬁt is made when revenue is greater than expenses. If less than 3,000 widgets are produced, the company will be operating in the red area of the graph, where E > R. The company is not making enough revenue to pay its expenses.

■ CHECK YOUR UNDERSTANDING If the company sells 2,900 widgets, is Wally’s Widget World operating above or below the breakeven point? What is the difference between revenue and expense?

You can give other examples of simultaneous linear equations and have the students solve them graphically and algebraically, as review.

CHECK YOUR UNDERSTANDING Answer (30,000, 210,000) Remind students that the dollar sign should not be included in the ordered pair. They should properly label the axes on their graphs to understand the y-coordinate represents a dollar amount.

EXAMPLE 5 Find the solution to Example 4 algebraically.

SOLUTION To ﬁnd the breakeven point, set the revenue and expense equations equal to each other. R=E Substitute.

2.50q = 1.10q + 4,200

Subtract 1.10q from each side.

1.40q = 4,200 1.40q 4,200 ______ = ______ 1.40 1.40

Divide each side by 1.40.

q = 3,000 The breakeven point occurs when the quantity produced, q, equals 3,000.

■ CHECK YOUR UNDERSTANDING Find the breakeven point for the expense function, E = 5.00q + 60,000, and the revenue function, R = 7.00q.

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Fixed and Variable Expenses

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Applications An economist is an expert who will know tomorrow why the things he predicted yesterday didn’t happen today. Laurence J. Peter, Professor TEACH

1. Interpret the quote according to what you have learned in this chapter. See margin.

Units in Answers Point out to students that correlation coefﬁcients have no units. Always require them to give answers, especially answers that are slopes, with the units. The more they hear it, read it, and say it, the more comfortable they will be with using the units. In science classes, they are encouraged/required to give units in answers.

ANSWERS 1. Many variables are considered when making business decisions. Although economists use math models to try to predict what will happen, a change in any variable can cause a change in the expected result.

2. The Gidget Widget Corporation produces widgets. The ﬁxed expenses are $65,210, and the variable expenses are $4.22 per widget. Express the expense function algebraically. E = 4.22q + 65,210 3. A corporation produces mini-widgets. The variable expenses are $1.24 per mini-widget, and the ﬁxed expenses are $142,900. a. How much does it cost to produce 1 mini-widget? $142,901.24 b. How much does it cost to produce 20,000 mini-widgets? $167,700 c. Express the expense function algebraically. E = 1.24q + 142,900 d. What is the slope of the expense function? 1.24 e. If the slope is interpreted as a rate, give the units that would be used. dollars per mini-widget 4. The expense function for the Wonder Widget is E = 4.14q + 55,789. a. What is the ﬁxed cost in the expense function? $55,789 b. What is the cost of producing 500 Wonder Widgets? $57,859 c. What is the average cost per widget of producing 500 Wonder Widgets? Round to the nearest cent. $115.72 d. What is the total cost of producing 600 Wonder Widgets? $58,273 e. What is the average cost per widget of producing 600 Wonder Widgets? Round to the nearest cent. $97.12 f. As the number of widgets increased from 500 to 600, did the average expense per widget increase or decrease? decrease g. What is the average cost per widget of producing 10,000 Wonder Widgets? Round to the nearest cent. $9.72 5. The Royal Ranch Pool Supply Corporation manufactures chlorine test kits. The kits have an expense equation of E = 5.15q + 23,500. What is the average cost per kit of producing 3,000 test kits? Round to the nearest cent. $12.98 6. The ﬁxed costs of producing a Wild Widget are $34,000. The variable costs are $5.00 per widget. What is the average cost per widget of producing 7,000 Wild Widgets? Round to the nearest cent. $9.86 7. Wanda’s Widgets used market surveys and linear regression to develop a demand function based on the wholesale price. The demand function is q = –140p + 9,000. The expense function is E = 2.00q + 16,000. a. Express the expense function in terms of p. E = –280p + 34,000 b. At a price of $10.00, how many widgets are demanded? 7,600 c. How much does it cost to produce the number of widgets from part b? $31,200

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8. Wind Up Corporation manufactures widgets. The monthly expense equation is E = 3.20q + 56,000. They plan to sell the widgets to retailers at a wholesale price of $6.00 each. How many widgets must be sold to reach the breakeven point? 20,000 9. The Lerneg Corporation computed its monthly expense equation as E = 11.00q + 76,000. Its products will be sold to retailers at a wholesale price of $20.00 each. How many items must be sold to reach the breakeven point? Round to the nearest integer. 8,444. 10. Solve Exercise 9 using the graph intersection feature on your graphing calculator. Are the answers equivalent? Yes; see margin.

12. Examine the graph of expense and revenue. a. What is the breakeven point? (A, W ) b. If quantity C is sold and C < A, is there a proﬁt or a loss? Explain. See margin. c. If quantity D is sold and D > A, is there a proﬁt or a loss? Explain. See margin. d. The y-intercept of the expense function is Z. Interpret what the company is doing if it operates at the point (0, Z).

dollars

11. Variable costs of producing widgets account for the cost of gas required to deliver the widgets to retailers. A widget producer ﬁnds the average cost of gas per widget. The expense equation was recently adjusted from E = 4.55q + 69,000 to E = 4.98q + 69,000 in response to the increase in gas prices. a. Find the increase in the average cost per widget. $0.43 b. If the widgets are sold to retailers for $8.00 each, ﬁnd the breakeven point prior to the adjustment in the expense function. (20,000, 160,000) c. After the gas increase, the company raised its wholesale cost from $8 to $8.50. Find the breakeven point after the adjustment in the expense function. Round to the nearest integer. (19,602, 166,618)

R E

W Z A

See margin.

TEACH Domains and Breakeven Points Constantly remind students what the domains before and after the breakeven points depict. For visual learners, this is crucial, and for non-visual learners, it may be difﬁcult.

ANSWERS 10.

Intersection X=8444.4444

Y=168888.89

12b. There is a loss because expenses are greater than revenue at any quantity less than A. 12c. There is a proﬁt because revenue is greater than expenses at any quantity greater than A. 12d. If no items are sold, the company still has to pay Z dollars for the ﬁxed expenses. 6.21q + 125.000 13a. _______________ q 13b.

q

13. Billy invented an innovative baseball batting glove he named the Nokee and made his own TV infomercial to sell it. The expense function for the Nokee is E = 6.21q + 125,000. The Nokee sells for $19.95. a. Represent the average expense A for one Nokee algebraically. See margin. b. Set your calculator viewing window to show x-values between 0 and 1,000, and y-values from 0 to 2,000. Let x represent q and let y represent A. Graph the average expense function. See margin. c. Is the average expense function linear? no d. Is the average expense function increasing or decreasing as q increases? decreasing e. If only one Nokee is produced, what is the average cost per Nokee to the nearest cent? $125,006.21 f. If 100,000 Nokees are produced, what is the average cost per Nokee to the nearest cent? $7.46 14. Lorne has determined the ﬁxed cost of producing his new invention is N dollars. The variable cost is $10.75 per item. What is the average N cost per item of producing W items? 10.75W + ___ W

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Fixed and Variable Expenses

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Money often costs too much. Ralph Waldo Emerson, Poet

2-5 Graphs of Expense and Revenue Functions Key Terms

Objectives

• • • •

• Write, graph, and interpret the expense function.

nonlinear function second-degree equation quadratic equation parabola

• • • •

leading coefﬁcient maximum value vertex of a parabola axis of symmetry

• Write, graph, and interpret the revenue function.

How can expense and revenue be graphed?

• Identify the points of intersection of the expense and revenue functions. • Identify breakeven points, and explain them in the context of the problem. EXAMINE THE QUESTION A picture is worth 1,000 words. The graphs of expense and revenue equations yield pictorial insight into how expense and revenue interact.

CLASS DISCUSSION How does price contribute to consumer demand? Name some other factors that might also play a role in the quantity of a product consumers purchase. Why does a non-vertical line have slope but a nonlinear function does not? Students often confuse –ax 2 with (–ax)2. Highlight the difference and explain how the absence of parentheses changes the result.

86

Chapter 2

49657_02_ch02_p062-113.indd 86

The total expense for the production of a certain item is the amount of money it costs to manufacture and place it on the market. One contributor to consumer demand is the price at which an item is sold. Expense relies on the quantity produced and demand relies on price, so the expense function can be written in terms of price. Recall that revenue is the total amount a company collects from the sale of a product or service. Revenue depends on the demand for a product, which is a function of the price of the product. The relationship between price, demand, expense, and revenue can be better understood when the functions are graphed on a coordinate plane. Both the demand and expense functions are linear. However, as you will see, revenue is a nonlinear function when it is expressed in terms of price. That is, the graph of the revenue function is not a straight line. The revenue function has a variable raised to an exponent of 2, so it is a second-degree equation and is known as a quadratic equation. A quadratic equation can be written in the form y = ax2 + bx + c where a, b, and c are real numbers and a ≠ 0 The graph of a quadratic equation is called a parabola. In the revenue graph pictured, the horizontal axis represents the price of an item, and the vertical axis represents the revenue. If the leading coefﬁcient, a, is positive, then the parabola opens upward.

Parabola with a Positive Leading Coefficient

Modeling a Business

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If the leading coefﬁcient is negative, then the parabola opens downward. The downward parabola models the revenue function. In the graph, the parabola reaches a maximum value at its peak. This point is the vertex of the parabola and yields the price at which revenue can be maximized. The axis of symmetry is a vertical line that can be drawn through the vertex of the parabola so that the dissected parts of the parabola are mirror images of each other. The point on the horizontal axis through which the axis of symmeb try passes is determined by calculating – ___. 2a

Parabola with a Negative Leading Coefficient

Maximum

Axis of symmetry

Skills and Strategies The graphical relationship between revenue and expense reveals much information helpful to a business.

EXAMPLE 1 A particular item in the Picasso Paints product line costs $7.00 each to manufacture. The ﬁxed costs are $28,000. The demand function is q = –500p + 30,000 where q is the quantity the public will buy given the price, p. Graph the expense function in terms of price on the coordinate plane. The expense function is E = 7.00q + 28,000. Substitute for q to ﬁnd the expense function in terms of price.

SOLUTION

Expense function.

E = 7.00q + 28,000

Substitute.

E = 7.00(–500p + 30,000) + 28,000

Use the Distributive property.

E = –3,500p + 210,000 + 28,000

Simplify.

E = –3,500p + 238,000

The horizontal axis represents price, and the vertical axis represents expense. Both variables must be greater than 0, so the graph is in the ﬁrst quadrant. To determine a viewing window, ﬁnd the points where the expense function intersects the vertical and horizontal axes. Neither p nor E can be 0 because both a price of 0 and an expense of 0 would be meaningless in this situation. But, you can use p = 0 and E = 0 to determine an appropriate viewing window. To ﬁnd the vertical axis intercept, let p = 0 and solve for E. Let p = 0.

E = –3,500p + 238,000 E = –3,500(0) + 238,000 E = 238,000

To ﬁnd the horizontal axis intercept, let E = 0 and solve for p. Let E = 0.

Review a parabola and its related equation. Have students identify what is different in this equation from linear equations. Discuss how the leading coefﬁcient, in this case the coefﬁcient on the x 2 term, determines if the parabola opens upward or downward. Relate this concept to the leading coefﬁcient of a linear equation, which is the coefﬁcient on the x term and determines if the lines go up and to the right or down and to the right. In other words, the leading coefﬁcient determines if the line has a positive slope or a negative slope.

EXAMPLE 1 It is very important that students understand the concept of writing a function in terms of a stated variable. In Example 1, they must ﬁrst construct an expense function in terms of the variable q. Then, they must use the given demand function and substitute the expression for q so that the expense function is now in terms of the price p. This skill will be needed throughout the chapter.

0 = –3,500p + 238,000 3,500p = 238,000 p = 68

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TEACH

Graphs of Expense and Revenue Functions

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CHECK YOUR UNDERSTANDING Answer E = –1,000p + 220,000 Viewing window used: Xmin = 0 Ymin = 200,000 Xmax = 15 Ymax = 250,000 Xscl = 1 Yscl = 1,000

dollars Use the intercepts to deter238,000 mine the size of the viewing window. In order to get a full picture, use window values that are slightly greater than the intercepts. Choose an appropriate scale for each axis. In this case, set the horizontal axis at 0 to 70 with a scale of 68 10 and the vertical axis at 0 to 240,000 with the scale at 50,000. Enter the function and graph.

p

■ CHECK YOUR UNDERSTANDING Students must ﬁrst create the expense function in terms of q; E = 5q + 20,000. Then, substitute the expression for q given in the problem statement. This yields an expense function in terms of the price p. The horizontal intercept is 220. This can help you select viewing window values of 0 to 250 with a scale of 50. The vertical intercept is 220,000.

EXAMPLE 2

CHECK YOUR UNDERSTANDING Answer $437,500

Revenue is the product of the price and quantity, or R = pq. The quantity, q, is expressed in terms of the price, so the expression can be substituted into the revenue equation.

Revenue equation

R = pq

Substitute.

R = p(–500p + 30,000)

Distribute.

R = p(–500p) + p(30,000)

Simplify.

R = –500p2 + 30,000p

The revenue function is a quadratic equation in the form y = ax2 +bx + c where a = –500, b = 30,000, and c = 0. Because the leading coefﬁcient is negative, the graph is a parabola that opens downward.

Determine the revenue if the price per item is set at $25.00.

EXAMPLE 3

EXAMPLE 3 The revenue equation is a parabola, so students may need guidance in selecting an appropriate viewing window. Finding the axis of symmetry and the vertex will help set the vertical dimensions.

49657_02_ch02_p062-113.indd 88

What is the revenue equation for the Picasso Paints product? Write the revenue equation in terms of the price.

■ CHECK YOUR UNDERSTANDING

This answer can also be found by determining the demand at that price and then multiplying that quantity by $25.

Chapter 2

EXAMPLE 2

SOLUTION

Students often confuse revenue and proﬁt. Emphasize that revenue is merely the amount of money that the sale of an item brings in. It is therefore a function of the price of the item and the quantity sold. Proﬁt is calculated taking into account the costs of making and selling the item.

88

An electronics company manufactures earphones for portable music devices. Each earphone costs $5 to manufacture. Fixed costs are $20,000. The demand function is q = –200p + 40,000. Write the expense function in terms of q and determine a suitable viewing window for that function. Graph the expense function.

Graph the revenue equation on a coordinate plane.

SOLUTION

The graph opens downward. It has a maximum height at the vertex. Here b = 30,000 and a = –500. –30,000 ________ –30,000 –b ________ ___ = = 30 = 2a

2(–500)

–1,000

Modeling a Business

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Revenue equation

R = –500p2 + 30,000p

Substitute 30 for p.

R = –500(30)2 + 30,000(30)

Simplify.

R = 450,000

maximum

dollars

The axis of symmetry passes through 30 on the horizontal axis. To determine the height of the parabola at 30, evaluate the parabola when p = 30.

When the price is set at $30, the maximum revenue is $450,000. The vertical axis must be as high as the maximum value. For the viewing window, use a number slightly greater than the maximum value, such as 500,000, so you can get the entire picture of the graph. Use the same horizontal values on the axis as used for the expense function.

p

CHECK YOUR UNDERSTANDING Answer Because $28 is closer to $30 than $40, it will yield a higher point on the parabola.

■ CHECK YOUR UNDERSTANDING

Students could test this statement by substitution but should be encouraged to think through the question before doing so. The very structure of a downward parabola yields a higher y-value as x approaches the axis of symmetry.

Use the graph in Example 3. Which price would yield the higher revenue, $28 or $40?

EXAMPLE 4 The revenue and expense functions are graphed on the same set of axes. The points of intersection are labeled A and B. Explain what is happening at those two points.

EXAMPLE 4 A and B are points of intersection. Ask students what is true about the y-value in each of the functions when evaluated at the x-values of points A and B. Once equality is establish, discuss what it means for the y-values to be the same for both functions at the points of intersection in the context of expense and revenue.

SOLUTION

B 8.08

58.92

CHECK YOUR UNDERSTANDING

p

Answer At both prices,

■ CHECK YOUR

UNDERSTANDING Why is using the prices of $7.50 and $61.00 not in the best interest of the company?

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49657_02_ch02_p062-113.indd Sec12:89

A

© IOFOTO 2009/USED UNDER LICENSE FROM SHUTTERSTOCK.COM

Before intersection point A, the expenses are greater than the revenue. The ﬁrst breakeven price is approximately at $8.08. The second is approximately at $58.92. From point A to point B, the revenue is greater than the expenses. To the right of point B, the expenses again are greater than the revenue.

dollars

Where the revenue and the expense functions intersect, revenue and expenses are equal. These are breakeven points. Notice there are two such points.

revenue is less than expenses. The price of $7.50 is too low for the incoming revenue to make up for the outgoing expenses. The price of $61 may be too high for consumers and would decrease the demand for the product. Therefore, with fewer items being sold, the revenue will be less.

Graphs of Expense and Revenue Functions

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Applications

Money often costs too much. Ralph Waldo Emerson, Poet TEACH

1. How might the quote apply to what you have learned? See margin.

Exercises 2 and 3 Students are introduced to a new product in each and given information necessary for creating an expense function. The independent variable for the expense function will be the quantity, q. Parts a–g of each question guide the students through the steps to graph the functions.

ANSWERS 1. Emerson cautions to be aware of the fact that the expenses involved in making revenue are often more than the revenue itself. 2c. The intercepts are (0, 10,000) and (10, 0), so the viewing window can be set from 0 to 12 with a scale of 1 for the horizontal axis and 0 to 12,000 for the vertical axis with a scale of 100. 2f. The maximum revenue of $18,062.50 is achieved when the price is $4.25; see additional answers for graph. 2g. (1.21, 8,794.36) and (8.29, 1,705.64); $1.21 and $8.29; see additional answers for graph. 3c. The intercepts are (0, 140,000) and (56, 0), so the viewing window can be set from 0 to 60 for the horizontal axis with a scale of 5 and 0 to 150,000 for the vertical axis with a scale of 10,000. 3f. The maximum revenue of $200,000 is achieved when the price is set at $20.00; see additional answers for graph.

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2. Rich and Betsy Cuik started a small business. They manufacture a microwavable coffee-to-go cup called Cuik Cuppa Coffee. It contains spring water and ground coffee beans in a tea-bag-like pouch. Each cup costs the company $1.00 to manufacture. The ﬁxed costs for this product line are $1,500. Rich and Betsy have determined the demand function to be q = –1,000p + 8,500, where p is the price for each cup. a. Write the expense equation in terms of the demand, q. E = 1.00q + 1,500 b. Express the expense equation found in part a in terms of the price, p. E = –1,000p + 10,000 c. Determine a viewing window on a graphing calculator for the expense function. Justify your answer. See margin. d. Draw and label the graph of the expense function. See additional answers. e. Write the revenue function in terms of the price. R = –1,000p2 + 8,500p f. Graph the revenue function in a suitable viewing window. What price will yield the maximum revenue? What is the revenue at that price? Round both answers to the nearest cent. See margin. g. Graph the revenue and expense functions on the same coordinate plane. Identify the points of intersection using a graphing calculator. Round your answers to the nearest cent. Identify the price at the breakeven points. See margin. 3. Orange-U-Happy is an orange-scented cleaning product that is manufactured in disposable cloth pads. Each box of 100 pads costs $5 to manufacture. The ﬁxed costs for Orange-U-Happy are $40,000. The research development group of the company has determined the demand function to be q = –500p + 20,000, where p is the price for each box. a. Write the expense equation in terms of the demand, q. E = 5q + 40,000 b. Express the expense function in terms of the price, p. E = –2,500p + 140,000 c. Determine a viewing window on a graphing calculator for the expense function. Justify your answer. See margin. d. Draw and label the graph of the expense function. See additional answers. e. Write the revenue function in terms of the price. R = –500p2 + 20,000p f. Graph the revenue function in a suitable viewing window. What price will yield the maximum revenue? What is the revenue at that price? Round answers to the nearest cent. See margin. g. Graph the revenue and expense functions on the same coordinate plane. Identify the points of intersection using a graphing calculator, and name the breakeven points. Round to the nearest cent. Identify the price at the breakeven points. (7.46, 121,354.02) and (37.54, 46,145.98)/$7.46 and $37.54; see additional answers for graph.

Modeling a Business

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Risk comes from not knowing what you’re doing. Warren Buffet, Businessman

2-6

Breakeven Analysis Key Terms

Objectives

• •

• Determine the breakeven prices and amounts using technology or algebra.

zero net difference quadratic formula

What happens when revenue equals expense? When the revenue function is a quadratic dollars 500,000 equation, you can determine the values at which the expense and revenue functions are equal. Graph the revenue and expense functions on the same coordinate plane. The functions intersect at the values at which the expense and revenue functions are equal (blue points), or the breakeven points. The difference between expense and revenue equals zero, which is a zero net difference. Notice that the prices on the revenue function between the breakeven points result 0 in revenue greater than expense. Prices beyond Revenue less the breakeven points result in revenue less than than expenses expense. Decision makers must examine the breakeven points carefully to set appropriate prices to yield maximum revenue. Breakeven points may be found algebraically or by using a graphing calculator. To ﬁnd breakeven points algebraically, set the expense and revenue functions equal to each other. Rewrite the resulting equation with 0 on one side of the equal sign and all the remaining terms on the other side. This gives a quadratic equation in the form ax2 + bx + c = 0, where a ≠ 0. The quadratic formula can then be used to solve this quadratic equation. The quadratic formula is _______

–b ± √b2 – 4ac x = _____________ where ± (read plus or minus) indicates 2a the two solutions ________

–b + √b – 4ac x = ______________ and 2

2a

Revenue greater than expenses

Breakeven points price 100

EXAMINE THE QUESTION This lesson examines what happens at the intersection of those functions. It might be helpful if you ask “What is true about the intersection of a street and an avenue?” Elicit responses that refer to the fact that the pavement at the intersection is shared by both the street and the avenue. Use this as a springboard for the discussion about intersection on graphs.

________

–b – √b2 – 4ac x = _____________ 2a

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Breakeven Analysis

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CLASS DISCUSSION

Skills and Strategies

Why wouldn’t a company set the price of an item at the breakeven point price? Are there always two breakeven points? Could there be a situation where there are more than two? Less than two?

Breakeven analysis can be used to make product price-setting decisions. Calculation and interpretation must be done during breakeven analysis.

EXAMPLE 1

Ask students to describe situations where a graph might also be a quadratic. What are you ﬁnding when you solve a quadratic equation?

TEACH Use the labeled graph from page 91 to be certain students can interpret these graphs when they create them in this lesson. If your students do not remember the quadratic formula and want to know where it comes from, show the process of ﬁnding solutions by completing the square for the equation y = ax 2 + bx + c.

EXAMPLE 1

Determine the prices at the breakeven points for the Picasso Paints product in Lesson 2-5. The expense function is E = –3,500p + 238,000, and the revenue function is R = –500p2 + 30,000p.

SOLUTION

The breakeven point occurs when the expense and revenue functions intersect. There are two ways to determine the breakeven point—using technology or algebra. If a graphing calculator is used, let x represent the price, and express the equations in terms of x. Enter each function into the calculator, and graph them both using a suitable viewing window. Find the intersection points. Because the calculator will display answers to many decimal places, round the intersection points to the nearest cent. The ﬁgures below show two graphing calculator screens after the Intersect feature has been used. Shown below on the left is the leftmost breakeven point; the price is at $8.08. On the right is the rightmost breakeven point; the price is $58.92.

Make sure that students are comfortable with both methods of ﬁnding the breakeven points. Discuss the role that rounding plays in creating a possible discrepancy in the vertical axis values at the breakeven points. Some students might ﬁnd the quadratic formula more accessible if________ it is stated as b _________ √b2 − 4ac ___ . Here, ± x=− 2a 2a students can see that the same amount is added to and subtracted from the axis of symmetry x-value, yielding roots that are symmetrical with respect to the axis of symmetry.

Intersection

X=8.0785524

Y=209725.07

Intersection

X=58.921448

Y=31774.933

The breakeven point can also be found algebraically. Set the expense and revenue functions equal to each other. Rewrite the resulting equation with 0 on one side and all the remaining terms on the other side. The quadratic equation is usually written with the leading coefﬁcient positive. E=R Set the expressions equal.

–3,500p + 238,000 = –500p 2 + 30,000p

Rewrite to set equal to 0.

+500p2 – 30,000p = +500p2 – 30,000p

500p2 – 33,500p + 238,000 = 0 In the quadratic equation 500p2 – 33,500p + 238,000 = 0, a = 500, b = –33,500, and c = 238,000. Because the equation is written in terms of p, use p instead of x in the quadratic formula. Substitute values for a, b, and c, and calculate.

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________

CHECK YOUR UNDERSTANDING

–b ± √b – 4ac x = ______________ 2

Quadratic formula

2a

Use + in the formula.

Answer The breakeven

___________________________

prices using both the quadratic formula method and the graphing calculator are $8.33 and $25. The former has been rounded to two decimal places.

√ p = _________________________________________ ≈ 58.92 –(–33,500) +

(–33,500)2 – 4(500)(238,000) 2(500)

Use – in the formula. p=

___________________________

–(–33,500) – √(–33,500)2 – 4(500)(238,000) ________________________________________

≈ 8.08 2(500) The prices at the breakeven points are $8.08 and $58.92.

EXTEND YOUR UNDERSTANDING Answer The actual

■ CHECK YOUR UNDERSTANDING

breakeven values have non-terminating decimal parts, so using the rounded values will not yield exact expense and revenue values. The expense and revenue values are approximately equal when p = 8.08 and p = 58.92.

The expense function for a particular product is E = –2,000p + 125,000. The revenue function for that product is R = –600p2 + 18,000p. Determine the prices at the breakeven points for this product both algebraically and graphically.

EXAMPLE 2

■ EXTEND YOUR UNDERSTANDING Knowing that the two breakeven prices have been rounded to the nearest cent, what would you expect when each is substituted into the expense and revenue equations?

Example 2 follows from the discussion in Extend Your Understanding. Notice that the expense and revenue values are approximately equal due to rounding.

EXAMPLE 2

CHECK YOUR UNDERSTANDING

Determine the revenue and expense for the Picasso Paints product at the breakeven points found in Example 1.

Answer The error could have been improved by using more accurate prices with more decimal places in the calculation. But, for this purpose, it is acceptable to round to the nearest cent.

The y-value of the breakeven point will be both the revenue and expense values. These values can be determined by substituting the values of p into the expense and revenue equations.

SOLUTION

Substitute 8.08 for p.

E = –3,500(8.08) + 238,000 = 209,720 R = –500(8.08)2 + 30,000(8.08) = 209,756.80

Substitute 58.92 for p.

E = –3,500(58.92) + 238,000 = 31,780 R = –500(58.92)2 + 30,000(58.92) = 31,816.80

When the breakeven price is approximately $8.08, the expense and revenue values are each close to $209,750. When the breakeven price is approximately $58.92, the expense and revenue values are each approximately $31,800. Because of rounding, the expense and revenue are not equivalent. They are, however, approximately equal.

■ CHECK YOUR UNDERSTANDING How could you have improved on the error when calculating the expense and revenue values?

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Breakeven Analysis

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EXAMPLE 3

EXAMPLE 3

Carefully walk the students through the solution as stated here. Without a good understanding of the algebraic manipulations, students will not be able to create the spreadsheet formulas. Impress upon students that there are other acceptable correct solutions in spreadsheet form that will yield the same result.

Use a spreadsheet to determine the breakeven price for the Picasso Paints product.

SOLUTION Develop general expense and revenue equations so the spreadsheet can calculate the breakeven prices regardless of the situations. The expense function is E = Vp + F, where p equals price, V equals variable costs, and F equals ﬁxed costs. The revenue function is R = Ap2 + Bp where p equals price and A and B are values speciﬁc to the particular situation. Set the expense equation equal to the revenue equation. Subtract Vp + F from both sides.

CHECK YOUR UNDERSTANDING

E=R Vp + F = Ap2 + Bp –Vp – F = –Vp – F 0 = Ap2 + Bp – Vp – F 0 = Ap2 + ( B – V )p – F

Answer Because the value

Combine like terms.

in B11 is the same as the value in B7, the formula in B11 is =B7. The value in B12 is the difference between B and V. Therefore, the formula in B12 is =(B8–B3). The value in B13 is the negative value in B4. The formula in B13 is =–B4.

This is a quadratic equation where a = A, b = (B – V), and c = –F. Depending on the spreadsheet software, cell formulas may vary. The Picasso Paints functions are E = –3,500p + 238,000 and R = –500p2 + 30,000.

If you want to stress to students that they are taking the opposite of the value in B4, you can have them write the formula for B13 as =–1*B4.

The values of V and F from the expense function are entered into cells B3 and B4. The values of A and B from the revenue function are entered into cells B7 and B8. Cells B15 and B16 are the breakeven prices calculated using the quadratic formula. For B15 the formula is =(–B12+SQRT(B12ˆ2–4*B11*B13))/(2*B11). For B16 the formula is =(–B12–SQRT(B12ˆ2–4*B11*B13))/(2*B11).

1 2 3

A Breakeven Calculator

B

The expense equation has the form Vp + F.

–3,500p + 238,000 –3,500 238,000

Enter the value of V in cell B3.

4

Enter the value of F in cell B4.

5 6 7 8 9 10

The revenue equation has the form Ap 2 + Bp. –500p 2 + 30,000p Enter the value of A in cell B7. Enter the value of B in cell B8. Solve the quadratic equation.

–500 30,000

–500p 2 + 33,500p + –238,000

11

where a =

12

b=

33,500

13

c=

–238,000

14 15

The price at the ﬁrst breakeven point is

16

The price at the second breakeven point is

–500

$8.08 $58.92

■ CHECK YOUR UNDERSTANDING What cell formulas were used to identify the a, b, and c values shown in B11, B12, and B13?

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Applications

Risk comes from not knowing what you’re doing. Warren Buffet, Businessman TEACH

1. How might the quote apply to what you have learned? See margin. 2. A manufacturer has determined that the combined ﬁxed and variable expenses for the production and sale of 500,000 items are $10,000,000. What is the price at the breakeven point for this item? $20 3. A supplier of school kits has determined that the combined ﬁxed and variable expenses to market and sell G kits is W. a. What expression models the price of a school kit at the G breakeven point? ___ W b. Suppose a new marketing manager joined the company and determined that the combined ﬁxed and variable expenses would only be 80% of the cost if the supplier sold twice as many kits. Write an expression for the price of a kit at the breakeven point 2G using the new marketing manager’s business model. _____

Exercises 2 and 3 These questions are related and should be assigned together. Exercise 2 offers a purely numerical situation and Exercise 3 offers a related algebraic situation.

Exercises 5 and 6 These questions offer students numerical information about the relationship between revenue and expense. They are asked to use this data (rather than equations) to sketch graphs.

0.8W

4. A jewelry manufacturer has determined the expense equation for ANSWERS necklaces to be E = 1,250q + 800,000, where q is the quantity 1. Warren Buffet speaks to demanded. At a particular price, the breakeven revenue is $2,600,000. the need to be knowledgeable in order to a. What is the quantity demanded at the breakeven point? 1,440 necklaces minimize or eliminate b. If the breakeven revenue changes to 3.5 million, will the quanrisk. With the knowltity demanded have increased or decreased? Explain. See margin. edge of the prices and 5. A manufacturer determines that a product will reach the breakeven point if sold at either $80 or $150. At $80, the expense and revenue values are both $300,000. At $150, the expense and revenue values are both $100,000. On graph paper, graph possible revenue and expense functions that depict this situation. Circle the breakeven points. Answers vary. 6. iSports is considering producing a line of baseball caps with wireless cellphone earpieces attached. The breakeven point occurs when the price of a cap is $170 or $350. At $170, the expense and revenue values are both $2,600,000. At $350, the expense and revenue values are both $900,000. On graph paper, graph possible revenue and expense functions that depict this situation. Circle the breakeven points. Answers vary. 7. SeaShade produces beach umbrellas. The expense function is E = –19,000p + 6,300,000 and the revenue function is R = –1,000p2 + 155,000p. a. Graph the expense and revenue functions. Label the maximum and minimum values for each axis. Circle the breakeven points. See additional answers. b. Determine the prices at the breakeven points. See margin. c. Determine the revenue and expense amounts for each of the breakeven points. See margin.

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subsequent revenue at breakeven points, the risk of putting a product on the market can be minimized. 4b. Increased; substitute 3,500,000 into the expense equation. When you solve for p, the solution is 2,160, which is greater than 1,440. 7b. $51.38 and $122.62 7c. When p = $51.38, E = $5,323,780.00 and R = $5,323,995.60. When p = $122.62, E = $3,970,220.00 and R = $3,970,435.60.

Breakeven Analysis

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TEACH

Exercises 9 and 10 These questions are similar in structure and intent. Perhaps assign Exercise 9 to half the class and Exercise 10 to the other half. Have students from each group present the solutions to the other group.

ANSWERS 8c. When p = $210.27, E = $7,248,650.00 and R = $7,248,637.71. When p = $394.73, E = $6,326,350.00 and R = $6,326,337.71 9a. If the bookcase is set at $22.22 then the maximum revenue would be approximately $8,888.89. 10c. When p = $11.48, E = $9,556.00 and R = $9,558.71. When p = $35.40, E = $2,380.00 and R = $2,378.88. See additional answers for graph.

8. Where-R-U produces global positioning systems (GPS) that can be used in a car. The expense equation is E = –5,000p + $8,300,000, and the revenue equation is R = –100p2 + 55,500p. a. Graph the expense and revenue functions. Circle the breakeven points. See additional answers. b. Determine the prices at the breakeven points. Round to the nearest cent. $210.27 and $394.73 c. Determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent. See margin. 9. The student government at State College is selling inexpensive bookcases for dorm rooms to raise money for school activities. The expense function is E = –200p + 10,000 and the revenue function is R = –18p2 + 800p. a. At what price would the maximum revenue be reached? What would that maximum revenue be? Round to the nearest cent. See margin. b. Graph the expense and revenue functions. Circle the breakeven points. See additional answers. c. Determine the prices at the breakeven points. Round to the nearest cent. $13.08 and $42.48 d. Determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent. When p = 13.08,

E = 7,384.00 and R = 7,384.44. When p = 42.48, E = 1,504.00 and R = 1,502.09.

10. An electronics store is selling car chargers for cell phones. The expense function is E = –300p + 13,000 and the revenue function is R = –32p2 + 1,200p. a. At what price would the maximum revenue be reached? $18.75 b. What would that maximum revenue be? Round to the nearest cent. $11,250 c. Graph the expense and revenue functions. Circle the breakeven points. See margin. d. Determine the prices at the breakeven points. Round to the nearest cent. $11.48 and $35.40 e. Determine the revenue and expense amounts for each of the breakeven points. Round to the nearest cent. When p = 11.48, E = 9,556.00 and R = 9,558.71. When p = 35.40, E = 2,380.00 and R = 2,378.88.

Use the graph to answer Exercises 11–14. 11. At what price is the maximum proﬁt reached? $70,000

dollars 100,000

12. What are the breakeven prices? $40 and $80 13. Name two prices where the revenue is greater than the expenses.

Sample answers: $45 and $50

14. Name two prices where the revenue is less than the expenses. Sample answers: $35 and $150

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Nobody ever lost money taking a proﬁt. Bernard Baruch, Businessman

The Proﬁt Equation 2-7 Key Terms

Objectives

• •

• Determine a proﬁt equation given the expense and revenue equations.

proﬁt maximum proﬁt

How do revenue and expenses contribute to profit calculation?

• Determine the maximum proﬁt and the price at which that maximum is attained.

In the business world, two companies may produce and sell the same product. It is possible that both companies will have the same maximum revenue based on their individual revenue functions. However, other aspects of production may not be identical. Revenue is the amount of money a company makes from the sale of goods or services. While the maximum revenue may be the same for each product, the revenue and expense equations may be different. Figure A displays the graph of the revenue function for the product. Figure B displays the graph of the revenue function of a similar product produced by a different company. The graphs of the revenue functions yield the same maximum values but model different situations. 500,000

dollars

500,000

dollars

price 0

Figure A

70

500,000

dollars

Figure B

70

dollars

price

price 0

Figure C

0

70

Figure D

Figures C and D depict the same revenue functions but different expense functions. Figure C shows an expense function that is always greater than the revenue graph. Therefore, the company makes no money on the manufacture and sale of the product. The company cannot even reach a breakeven point because there are no breakeven points. In Figure D, the expense function is greater than the revenue function at certain prices, but there are also prices at which the opposite is true. A company would prefer Figure D to model the sale of its product because there are prices at which money coming in is greater than money spent. The money a company makes after expenses have been deducted from revenue is proﬁt. In Figure D, there are prices at which the company will make a proﬁt. In Figure C, no matter the price, the revenue is never enough to offset the expenses.

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Students have now explored the algebraic and graphical relationships between expense and revenue.

500,000

price 0

EXAMINE THE QUESTION

70

This chapter is about proﬁt and what factors contribute to proﬁt. Ask students for their deﬁnition of proﬁt. Then, ask them how revenue and expense contribute to proﬁt. Their response will pave the way for the lesson and act as a ﬁrst-step in answering the opening question.

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CLASS DISCUSSION Ask students for the names of different companies that produce the same product. Then, ask what aspects of the production might not be identical among the companies.

500,000

Describe a graph and a situation in which there is only one breakeven point. What does it mean for production and sales? Have students examine the graph. Ask them to identify three different prices at which proﬁt would be a small, medium, and large amount (not necessarily the maximum).

dollars

price

0

70

Figure E

500,000

Graphically, proﬁt is the vertical distance between the revenue and expense functions. In Figure E, the top of the vertical line segment (in purple) hits the revenue graph at $437,500 when the price is about $25. The bottom of the vertical line segment hits the expense graph at $150,500 at the same price. The vertical length of this segment is 437,500 – 150,500 = 287,000 and is the proﬁt the company makes when the price is about $25. P = R – E where P is proﬁt, R is revenue, and E is expenses

dollars

price

0

70

© MARCIN BALCERZAK 2009/USED UNDER LICENSE FROM SHUTTERSTOCK.COM

Figure F

In Figure F, several proﬁt line segments have been drawn for different prices. The longest segment would represent the greatest difference between revenue and expense at a given price. The greatest difference between revenue and expense denotes maximum proﬁt. It is difﬁcult to make a visual determination as to where the maximum proﬁt line might be drawn. Algebra and graphing must be used to make a more precise determination.

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Skills and Strategies TEACH

The revenue and expense functions are both functions of price, p, so, you can create a proﬁt function in terms of p.

Before you begin Example 1, choose a product that students can relate to such as cell phones. Name two different companies that manufacture cell phones and have students describe the differences in Figures A–D on page 97 by relating them to the cell phones and their manufacturers.

EXAMPLE 1 Determine the proﬁt equation for the Picasso Paints product in Lesson 2-5. The revenue and expense functions were R = –500p2 + 30,000p E = –3,500p + 238,000

SOLUTION

Proﬁt is the difference between revenue and expense.

EXAMPLE 1

P=R–E Substitute for R and E.

P = –500p2 + 30,000p – (–3,500p + 238,000)

Distribute.

P = –500p2 + 30,000p + 3,500p – 238,000

Combine like terms.

P = –500p2 + 33,500p – 238,000

Perhaps the most common error students make when generating the proﬁt equation is the failure to distribute the subtraction over both terms in the expense expression. Make sure that you alert them to the importance of this part of the computation.

The proﬁt equation is P = –500p2 + 33,500p – 238,000. It is a downward parabola because the leading coefﬁcient is negative.

■ CHECK YOUR UNDERSTANDING

CHECK YOUR UNDERSTANDING

Suppose that the revenue and expense functions are R = –350p2 + 18,000p and E = –1,500p + 199,000. Write the proﬁt equation.

Answer P = R – E P = –350p2 + 19,500p – 199,000

EXAMPLE 2

EXAMPLE 2

Replace the variable p with x and P with Y.

Use a graphing calculator to draw the graph of the proﬁt equation from Example 1. What is the maximum proﬁt?

CHECK YOUR UNDERSTANDING

SOLUTION

Enter the proﬁt equation dollars P = –500p +33,500p – 238,000 500,000 into a graphing calculator. Use the Maximum profit same viewing window you used for the graphs of the revenue and expense functions in Lesson 2-5. The maximum proﬁt is at the vertex of the parabola. Graphing calculators have a maximum feature that determines the value of the maximum point of a function. Let x represent the price, p. The maximum is 323,125 when 0 x = 33.50. 2

Answer The two points where the proﬁt function intercepts the horizontal axis indicate item prices at which no proﬁt is made. These prices are $8.07 and $58.93.

price 70

■ CHECK YOUR UNDERSTANDING Sketch the graph. Identify the points where zero proﬁt is made. Explain your reasoning.

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EXAMPLE 3

EXAMPLE 3

Students should maintain the same viewing window and all three graphs should be pictured in this window. Walk the students through each of the points mentioned in the problem statement and ask them to interpret the situation that is modeled at those points.

CHECK YOUR UNDERSTANDING

Graph the revenue, expense, and proﬁt functions on the same coordinate plane. Interpret the zero-proﬁt points, the maximum proﬁt, and how the functions relate to each other.

SOLUTION

The maximum proﬁt is at the vertex of the proﬁt graph. The price at this point yields the greatest difference between the revenue and expense functions. There are two zero-proﬁt points on the graph of the proﬁt function. No proﬁt is made at the selling price where expense is equal to revenue. Visually, this is the price at which the revenue and expense functions intersect.

Revenue

Answer No; the greatest difference between revenue and expense does not depend on the maximum revenue. You might wish to use a ruler to show that the maximum revenue distance and the maximum proﬁt are more than likely not the same.

EXAMPLE 4 To solve this problem algebraically, students need to ﬁrst determine the proﬁt quadratic and put it in standard form. Discuss where the maximum of a parabola always occurs (horizontal value is the axis of symmetry). Then, lead them through to the calculation of that maximum point. Although the question asks for the price at which the proﬁt is a maximum, the solution shows students how to use that price to determine the maximum proﬁt.

Maximum profit

Profit

Expenses Zero profit

■ CHECK YOUR UNDERSTANDING Must maximum proﬁt occur at the same price as the maximum revenue?

EXAMPLE 4 Algebraically, determine the price of the Picasso Paints product that yields the maximum proﬁt.

SOLUTION To determine the maximum proﬁt algebraically, recall that the maximum value occurs on the axis of symmetry. For the proﬁt function P = –500x2 + 33,500x – 238,000, a = –500, b = 33,500, and c = –238,000. The x- intercept of the axis of symmetry is determined –b by calculating ___. 2a –33,500 –33,500 ________ –b ________ ___ = = = 33.5 2a –1,000 2 (–500)

CHECK YOUR UNDERSTANDING

The axis of symmetry intercepts the horizontal axis at 33.5. To determine the height of the parabola at 33.5, evaluate the function when p = 33.5.

Answer Using P = –350p2 + 19,500p – 199,000, a maximum proﬁt of $72,607.14 is attained at a price of $27.86.

Quadratic equation

P = –500p2 + 33,500p – 238,000

Substitute 33.5 for p.

P = –500(33.5)2 + 33,500(33.5) – 238,000

Simplify.

P = 323,125

At $33.50, a maximum proﬁt of $323,125 is attained.

■ CHECK YOUR UNDERSTANDING Use the proﬁt function from Example 1 Check Your Understanding. Determine the price to the nearest cent that yields the maximum proﬁt.

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Applications

Nobody ever lost money taking a proﬁt. Bernard Baruch, Businessman

1. How might the quote apply to what has been outlined in this lesson? See margin.

ANSWERS

1. A person would lose money when the expense is greater than the revenue. A true proﬁt equation is found in the ﬁrst quadrant and is the difference between 2. 3. the revenue and the dollars dollars expense. If the expense is $100,000 $100,000 greater than the revenue, then there is no proﬁt. Therefore, no one would lose money if they take (make) a proﬁt! 2. The expenses are ﬁxed for this product. Regardless of price, the expenses are $500,000. A price price proﬁt is made when the 0 0 price is set approximately 100 100 between $20 and $60. 3. This situation is similar 4. 5. to that examined in Skills dollars dollars and Strategies. Expenses $100,000 $100,000 decrease as the price of the item increases. Proﬁt is made when the price is set approximately between $13 and $70. 4. The expenses are greater than the revenue at all possible prices. No proﬁt can be made from price price the sale of this item. 0 0 5. This situation is similar 100 100 to that examined in Skills and Strategies. Expenses decrease as the price of In Exercises 6–9, write the proﬁt function for the given expense the item increases. Proﬁt and revenue functions. is made when the price is set approximately 6. E = –20,000p + 90,000 between $7 and $18. R = –2,170p2 + 87,000p P = –2,170p2 + 107,000p – 90,000

Examine each of the graphs in Exercises 2–5. In each case, the blue graph represents the expense function and the black graph represents the revenue function. Describe the proﬁt situation in terms of the expense and revenue functions. See margin.

7. E = –6,500p + 300,000 R = –720p2 + 19,000p P = –720p2 + 25,500p – 300,000 8. E = –2,500p + 80,000 R = –330p2 + 9,000p

P = –330p2 + 11,500p – 80,000

9. E = –12,500p + 78,000 R = –1,450p2 + 55,000p P = –1,450p2 + 67,500p – 78,000

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TEACH

10. Examine the revenue (black) and expense (blue) functions. Estimate the price at the maximum proﬁt. Explain your reasoning. See margin.

Exercise 12 Encourage students to attempt at least one of these problems in two different ways (graphically and algebraically).

Exercises 13–16 These questions are similar in intent. You might want to randomly assign two problems to each student so that when reviewing them the next day, students can listen to their peers explain the individual work that was done.

800,000

dollars

11. The expense and revenue functions yield a proﬁt function, but the equation can represent price no proﬁt made for any price. 0 One of the proﬁt functions in 50 Exercises 6–9 models such a situation. a. Determine which proﬁt function models a no proﬁt situation. See margin. b. What does a proﬁt function look like when no proﬁt can be made? If expenses are always greater than the revenue, the proﬁt parabola will lie entirely below the horizontal axis.

ANSWERS 10. Approximately $26; it appears to have the greatest positive vertical distance from the revenue to the expense function. 11a. In Exercise 7 the expenses are always greater than the revenue. 12a. At $15.50, the maximum proﬁt would be $46,100. 12b. At $11.89, the maximum proﬁt would be $27,324.35. 12c. At $261.18, the maximum proﬁt would be $11,541,235.29.

12. Determine the maximum proﬁt and the price that would yield the maximum proﬁt for each. See margin. a. P = –400p2 + 12,400p – 50,000 b. P = –370p2 + 8,800p – 25,000 c. P = –170p2 + 88,800p – 55,000 13. Greenyard’s manufactures and sells yard furniture made out of recycled materials. It is considering making a lawn chair from recycled aluminum and fabric products. The expense and revenue functions are E = –1,850p + 800,000 and R = –100p2 + 20,000p. a. Determine the proﬁt function. P = –100p2 + 21,850p – 800,000 b. Determine the price, to the nearest cent, that yields the maximum proﬁt. $109.25 c. Determine the maximum proﬁt, to the nearest cent. $393,556.25 14. Mountaineer Products Incorporated manufactures mountain-bike accessories. It is considering making a new type of reﬂector for night biking. The expense and revenue functions are E = –450p + 90,000 and R = –185p2 + 9,000p. a. Determine the proﬁt function. P = –185p2 + 9,450p – 90,000 b. Determine the price, to the nearest cent, that yields the maximum proﬁt. $25.54 c. Determine the maximum proﬁt, to the nearest cent. $30,679.05 15. Business Bargains manufactures ofﬁce supplies. It is considering selling sticky-notes in the shape of the state in which they will be sold. The expense and revenue functions are E = –250p + 50,000 and R = –225p2 + 7,200p. a. Determine the proﬁt function. P = –225p2 + 7,450p – 50,000 b. Determine the price, to the nearest cent, that yields the maximum proﬁt. $16.56 c. Determine the maximum proﬁt, to the nearest cent. $11,669.44 16. FlipFlops manufactures beach sandals. Their expense and revenue functions are E = –300p + 32,000 and R = –275p2 + 6,500p. a. Determine the proﬁt function. P = –275p2 + 6,800p – 32,000 b. Determine the price, to the nearest cent, that yields the maximum proﬁt. $12.36 c. Determine the maximum proﬁt, to the nearest cent. $10,036.36

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All models are wrong. Some models are useful. George Box, Statistician and Quality Control Pioneer

Mathematically Modeling 2-8 a Business Key Terms

Objectives

• •

• Recognize the transitive property of dependence as it is used in a business model.

dependence transitive property of dependence

How can you mathematically model a start-up business? Statistics are necessary in making business decisions. The relationship between supply and demand; expense, revenue, and proﬁt; and breakeven points must be analyzed. All of the factors may be modeled together to assess business situations. Dependence is used in many contexts. • In sports, baseball fans depend on the manager of the team to lead the team to victory. In turn, the manager of the team depends on the players to work hard to succeed. • In politics, voters depend on their local elected ofﬁcials to represent them. Local elected ofﬁcials depend on state government ofﬁcials to give them the support they need to represent the voters. • When starting a business venture, expenses depend on the demanded quantity of the product. Demand depends on the price of the product. These are a few examples of dependence in daily life. In the ﬁrst example, if the fans depend on the manager and the manager depends on the players, the fans depend on the players as well. In the second example, if the voters depend on the local elected ofﬁcials and the local ofﬁcials depend on the state ofﬁcials, the voters depend upon the state ofﬁcials, too. Finally, in the last example, if expenses depend on quantity and quantity depends on price, expenses also depend on price. These are examples of the transitive property of dependence. If x depends on y and y depends on z, it follows that x depends on z. The determination of the price that yields the maximum proﬁt depends on a number of factors that precede it. Mathematical modeling using algebra is an illustration of the use of the transitive property in business.

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• Use multiple pieces of information, equations, and methodologies to model a new business.

EXAMINE THE QUESTION In order for this question to make sense, students need a good understanding of what a mathematical model is and when those models are used. Discuss some situations where real life situations are modeled mathematically in order to gain information that would otherwise be difﬁcult to attain.

CLASS DISCUSSION Name other situations that illustrate dependence. Have students use the situations identiﬁed in previous discussions and illustrate how they might be examples of the transitive property of dependence.

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TEACH Discuss with students how the models in this lesson differ from models used in math courses in earlier grades and how they are the same. Students will likely remember using models such as algebra tiles, fraction bars, and threedimensional models of geometric solids. They will not, however, usually think of graphing on a coordinate plane or writing and solving an equation as a model. Guide students to see that the fundamental use of a model is the same, whether it is an equation or a cube.

EXAMPLE 1 It is important to work through both methods with students so that they can see the nature of mathematical dependency in this situation.

Skills and Strategies Mathematically modeling a business situation helps you better understand the relationships between and among variables.

EXAMPLE 1 Determine the expense, E, for production of an item when the price, p, is $60 given E = 50q + 80,000 and q = 80p + 100,000. E depends on q, and q depends on p. To ﬁnd the value of E at a particular price, p, use substitution or express the expense equation directly in terms of price. Both methods illustrate the transitive property of dependence.

SOLUTION

The ﬁrst method uses the price given to ﬁnd a value for q that can be substituted into the equation for E. Method 1

Use the equation for q.

q = 80p + 100,000

Substitute 60 for p.

q = 80(60) + 100,000

Simplify.

q = 104,800

CHECK YOUR UNDERSTANDING

Use the equation for E.

E = 50q + 80,000

Substitute 104,800 for q.

E = 50(104,800) + 80,000

Answer Using either

Simplify.

E = 5,320,000

method, the expense is $5,248,000.

The second method substitutes the value of q in terms of p into the expense equation, and then uses the price given to ﬁnd the value of E.

EXTEND YOUR UNDERSTANDING

Method 2

Answer Once the value of z is known, then y can be calculated. With y calculated, then x can be determined. Once x is determined, it is substituted into the ﬁrst equation, and A can be evaluated.

Use the equation for E.

E = 50q + 80,000

Substitute 80p + 100,000 for q.

E = 50(80p + 100,000) + 80,000

Distribute.

E = 4,000p + 5,000,000 + 80,000

Calculate.

E = 4,000p + 5,080,000

Substitute 60 for p.

E = 4,000(60) + 5,080,000

Simplify.

E = 5,320,000

The expenses total $5,320,000.

■ CHECK YOUR UNDERSTANDING Determine the expense, E, for production of an item when the price, p, is $42 given E= 50q + 80,000 and q = 80p + 100,000.

■ EXTEND YOUR UNDERSTANDING Suppose A = 20x + 30, x = 30y + 40, and y = 40z + 50. Describe how the value of A depends on the value of z.

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EXAMPLE 2 A business model uses a summary analysis of the situation in terms of dependent variables. Examine the three graphs of a business situation for the production of widgets. The graphs depict numerical information that is needed to complete the summary analysis.

demand 150,000

Demand (40, 103,200)

Write the summary analysis in terms of the data presented in the graphs. The summary analysis should have the following format. In summary, to start this business, __?__ widgets should be manufactured. Each should be sold for $__?__. The breakeven point is reached at a price of $__?__ or $__?__, but a proﬁt is made at any price between those prices.

price 0

200 amount

900,000

(36.57, $806,666.67) (40, $800,000)

(14.83, 520,664.46)

At the selling price, there is revenue of $__?__ and expenses of $__?__, resulting in a proﬁt of $__?__.

Revenue

(40, $420,000)

(65.17, $319,335.54)

Expense price

0

100

SOLUTION

In summary, to start this business, 103,200 widprofit 900,000 gets should be manufactured. Each should be sold for $40. This is asking the price that will yield the maximum proﬁt and the quantity demanded at that price. (40, 380,000) The breakeven point is reached at a price of $14.83 or $65.17, but a proﬁt is made at any price Profit between those prices. price These amounts can be found at the 0 intersection points of the expense 100 and revenue functions. At the selling price, there is revenue of $800,000 and expenses of $420,000, resulting in a proﬁt of $380,000. Use the price that yields the maximum proﬁt, $40, in both the expense and proﬁt functions.

■ CHECK YOUR UNDERSTANDING Use the points labeled on the graphs to show that the maximum proﬁt at the selling price is the difference between the revenue and expense values at that price.

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EXAMPLE 2 Students may need assistance with this activity in determining what type of information each graph yields. Before attempting a solution, ask students to examine each graph and determine what information can be gained from it.

CHECK YOUR UNDERSTANDING Answer The maximum point on the proﬁt function is (40, $380,000). The expense at a price of $40 is $420,000, and the revenue at a price of $40 is $800,000. 800,000 – 420,000 = 380,000

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Applications

All models are wrong. Some models are useful. George Box, Statistician and Quality Control Pioneer TEACH

1. How might those words apply to what has been outlined in this lesson? See margin.

Exercise 2 This question is a prerequisite for all of the exercises to follow. It is very important that students have an understanding of the points identiﬁed on the graphs and what those points represent in the context of a start-up business. You might want to do this problem in class as a closure piece for the lesson.

2. Use the following graph to answer the questions below. Let Y1 = cost function and Y2 = revenue function.

A B

C

ANSWERS

0

1. Models simulate reality but they themselves are not reality. There are a ﬁxed number of variables that are used in models. In reality, there are many more variables that contribute to the outcome of a situation. So, while models can be wrong, they can still be useful. 2f. The maximum proﬁt occurs where there is the greatest vertical difference between the revenue and expenses functions. It appears that this might happen at an x value slightly to the left of E. 3. $125,000 $105,000 $40 $160 0

$100

$200

a. b. c. d. e.

D

E

FG

H

Explain the signiﬁcance of point (D, B). It is the ﬁrst breakeven point. Explain the signiﬁcance of point (E, A). It is the maximum revenue point. Explain the signiﬁcance of point (F, C). It is the second breakeven point. Explain the signiﬁcance of point (G, 0). It represents the price at which no revenue is made. Explain the signiﬁcance of point (H, 0). It represents the price at which there are no expenses.

f. Where do you think the maximum proﬁt might occur? See margin. 3. Draw a graph of cost function Y1 and revenue function Y2 that meet the following criteria. See margin. Maximum revenue: $125,000 Breakeven points: $40 and $160 Price at which no revenue is made: $200 Maximum expenses: $105,000 Use the following situation to answer Exercises 4–20. A company produces a security device known as Toejack. Toejack is a computer chip that parents attach between the toes of a child, so parents can track the child’s location at any time using an online system. The company has entered into an agreement with an Internet service provider, so the price of the chip will be low. Set up a demand function—a schedule of how many Toejacks would be demanded by the public at different prices. 4. As the price increases, what is expected to happen to the quantity demanded? decreases 5. The horizontal axis represents price, and the vertical axis represents quantity. Does the demand function have a positive or negative slope? Explain. The demand is decreasing; the slope is negative.

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6. The company decides to conduct a market research survey to determine the best price for the device. The variable p represents price, and q represents quantity demanded. The points are listed as (p, q). (14, 8,200), (11, 9,100), (16, 7,750), (16, 8,300), (14, 8,900), (17, 7,100), (13, 8,955), (11, 9,875), (11, 9,425), (18, 5,825) Make a scatterplot of the data. Does the data look like it has a linear form? The scatterplot has a linear form. See margin. 7. Find the regression equation. Remember the quantity demanded, q, is the dependent variable. Round the slope and y–intercept to the nearest hundredth. q = –423.61p + 14,315.94

TEACH

Exercises 4–20 These problems guide students through the modeling process. They must be assigned in their entirety since they are cumulative and sequential.

ANSWERS 6.

8. Is the linear regression line a good predictor? Explain. See margin. 9. Examine the data to see if there is any relationship between the price and the quantity demanded. Determine the correlation coefﬁcient between price and demand, rounded to nearest hundredth. Explain the signiﬁcance of the correlation coefﬁcient. See margin. 10. Fixed costs are $24,500, and variable costs are $6.12 per Toejack. Express expenses, E, as a function of q, the quantity produced. E = 6.12q + 24,500 11. Express the revenue, R, in terms of p and q. R = pq 12. Express the revenue, R, in terms of p. R = –423.61p2 + 14,315.94p 13. Use the transitive property of dependence to express expense, E, in terms of p. Round to the nearest hundredth. E = –2,592.49p + 112,113.55 14. Graph the expense and revenue functions. a. Determine an appropriate maximum horizontal-axis value. 50 b. Determine an appropriate maximum vertical-axis value. 130,000 c. Sketch the graphs of the expense and revenue functions.

8. Based on the correlation coefﬁcient of 0.92, the linear regression line is a good predictor. 9. r = –0.92 This is a signiﬁcant correlation because the coefﬁcient is close to –1. 19. a. 5,661; b. $19.96; c. $8.40; d. $31.52; e.$116,979.26; f. $60,367.45; g. $56,611.81.

See additional answers.

15. Determine the coordinates of the maximum point on the revenue graph. Round the coordinates to the nearest hundredth. (16.90, 120,952.14) 16. Determine the breakeven points. Round to the nearest hundredth. (8.40, 90,343.87) and (31.52, 30,403.76)

17. Express the proﬁt, P, in terms of p. P = –423.61p2 + 16,908.43p – 112,113.55 18. Graph the proﬁt function. Determine the coordinates of the maximum point of the proﬁt graph. At what price, p, is proﬁt maximized? Round to the nearest cent. This will be the price at which Toejack will sell! (19.96, 56,611.81). The proﬁt is maximized at a price of $19.96. See additional answers for graph.

19. Write the business summary statement by ﬁlling in the blanks. In summary, to start this business, __a.__ Toejacks should be manufactured. Each should be sold at $__b.__. The breakeven point is reached at a price of $__c.__ or $__d.__, but a proﬁt is made at any price between those prices. At the selling price, there is a revenue of $__e.__ and expenses of $__f.__, resulting in a proﬁt of $__g.__. See margin.

20. If shares of stock are sold with an initial value of $5 each, how many shares must be sold to get enough money to start the business? 12,074

2-8

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2

CHAPTER

Assessment

Real Numbers You Write the Story!!

You Write the Story Students’ answers will vary. Begin by having them focus on the shape of the curve. Why are the points at the beginning and at the end of the time cycle higher than those in the middle? Why are the mid-cycle points close to horizontal? Once they have established the nature of the graph, the article should be easy to write.

REALITY CHECK The Reality Check projects are a terriﬁc form of alternative assessment. They give students an additional avenue to show what they have learned, so their grades are not solely based on tests.

Examine the scatterplot below. It depicts the rate of failures reported for a single product. A product failure is when a product fails to do what it was manufactured to do. The diagram below is called a “bathtub curve.” Write a short newspaper-type article centered on this graph, based on a hypothetical situation you create. An electronic copy of the graph can be found at school.cengage.com/math/ﬁnancialalgebra. Copy and paste it into your article.

Product Failure Rate Over Time

Failure Rate

CHAPTER 2 ASSESSMENT REAL NUMBERS

0

The Reality Check Projects in this lesson ask students

0.5

1

1.5

2 2.5 Time in Years

3

3.5

4

4.5

Reality Check to do research, interview business workers, read the business section of a newspaper, and more. Each project is designed so that students can use what they have learned in this chapter as a springboard for further work.

1. Corporations pride themselves on the recognition of their logos. This project involves creating a game. Use the library or the Internet to ﬁnd logos of twelve popular brands. Look only for logos that do not include the corporate name. Paste the logos on a sheet of paper. With teacher approval, give the class the sheet of logos, and ask students to write the brand name next to each logo. 2. Use the Internet or library to ﬁnd more details on the failures of the three products mentioned at the beginning of the chapter: New Coke, the Edsel, and the Sony Betamax. Write a report on the speciﬁc reasons for each failure. 3. Find an example of a product not mentioned in this chapter that failed. Prepare a report detailing the story behind the product’s invention and its failure.

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4. Interview a local businessperson. Ask for examples of ﬁxed and variable expenses. Do not ask for amounts as that information is private. Make a comprehensive list. Also ask about the history of the business and how he or she became involved in it. 5. Trace the history of the portable, hand held calculator from inception to the present. Include brand names, sales ﬁgures, features, pictures, model names, etc. Prepare the ﬁndings in a report. 6. On January 1, 1962, the Beatles, looking for their ﬁrst recording contract, had an audition with Decca records. They were turned down, and legend has it that Decca said “ . . . guitar groups are on the way out.” Use the library and/or the Internet to research the international sales of Beatles recordings over the past 5 decades and any other ﬁnancially-related Beatles facts. Compare Decca’s decision to the product failures discussed in this chapter. What are the similarities and differences? Prepare the ﬁndings in a report. 7. How prevalent are brand names in society? With teacher approval, get two copies of the same newspaper or magazine. Divide the class into two teams. First, give the number of pages in that publication. Next, have each team predict the number of times a brand name will be used. Have each team read through the entire publication (teams may split up the reading) and count every brand name that is written in the publication. If a name is written multiple times, count every time it is written. Write in the newspapers or magazines to easily keep track. Compare ﬁndings, and if they differ, ﬁnd out where they differ. How did the ﬁndings compare to the predictions? 8. Search the Internet for three different websites that offer breakeven calculators. Compare and contrast them. What can one do that the other(s) can’t? Write up a recommendation for the use of one of the three calculators and justify the recommendation. 9. Look either in the Business section of a newspaper or online to identify three companies that claim to have made a proﬁt over a period of time. Research how they each deﬁne proﬁt, and state the monetary amounts that each is quoting as their proﬁt for the time period. 10. Many children set up lemonade stands in front of their homes when they were younger. What would the ﬁxed costs of a lemonade stand be? What would be the variable costs? At what price would a glass of lemonade be sold? Explain how to decide the price. Estimate revenues and proﬁt over a one-week period.

Dollars and Sense

Your Financial News Update

Go to www.cengage.com/school/math/ﬁnancialalgebra where you will ﬁnd a link to a website containing current issues about modeling a business.

Assessment

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Really? Really! REVISITED

In the Really? Really! feature at the beginning of the chapter, students were introduced to products that were market failures. Certainly, the manufacturers of these products envisioned market success. But, for numbers of reasons, each individual to the product, consumers just didn’t make the purchases. Here, students revisit one such product failure, the Edsel. The structure of this bar graph might be new to students and will need an explanation. The concepts of automobile depreciation and appreciation will be explored in detail in the Automobile chapter, but are worth mentioning here. Often some cars depreciate at the outset and then after numbers of years become a collector’s item. Once that happens, the demand for them increases and the value of the car appreciates. This appears to be the case with the Edsel. After completing the questions related to the graph, encourage students to follow up with the Internet research extension activity. Remind them of the role perception played in stock splits. Link that concept to how perception played a role in the sale of the Edsel.

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1958 Villager 9-Passenger Wagon 16000 14000 12000 Price (US$)

REALLY? REALLY! REVISITED

Ford Motor Company predicted 1958 Edsel sales to exceed 200,000 cars. Only 63,107 of the 1958 models were sold. The 1960 model saw only 2,846 cars sold. During the three years of production, approximately 110,847 Edsels were manufactured. Today, less than 6,000 of them remain. As a rare curiosity, they are prized by many car collectors and have grown in value. The following bar graph gives the values of a 1958 Edsel Villager 9-Passenger Station Wagon during the years 1999–2008. Because the value depends on the condition of the car, car appraisers rate the condition of a classic car. On this graph, Condition 2 is the best condition and Condition 6 is the poorest condition.

10000 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008

8000 6000 4000 2000 0

6

5

4 Condition

3

2

Source: Edsel.com, http://www.edsel.com/charts/e58vil9p.jpg

1. Give an approximate value for the 1958 Villager in a condition of 6 in 2008. $1,000 2. Give an approximate value for the 1958 Villager in a condition of 2 in 2008. $14,000 3. Approximate the difference between a 1958 Villager in a condition of 6 and in a condition of 2 in 2008. $13,000 4. Approximate the difference between a 1958 Villager in a condition of 2 in 1999 and in 2008. $7,000 Look at the general trend of the graph. If a failure is kept for long enough, it may prove to be a good decision! Go online and look for pictures of the original 1958 Edsel. Also ﬁnd photos of the 1958 Chevrolet, the 1958 Chrysler, and the 1958 Buick. Does the Edsel look very different from the other cars? Consumers thought it did! And that’s reality!

Modeling a Business

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Applications 1. Examine each scatterplot. Identify each as showing a positive correlation, a negative correlation, or no correlation. a.

c.

b.

positive

negative

none

2. If each set of bivariate data has a causal relationship, determine the explanatory and response variables for each set of data. a. number of hours spent reading and page number on which you are reading hours-explanatory; page number-response b. calories burned and number of minutes of exercising minutes exercised-explanatory; calories-response c. amount paid as income tax and the amount of a paycheck paycheck-explanatory; taxes-response d. pounds of hamburger used to make a meatloaf and number of people that can be fed from the meatloaf pounds of hamburgerexplanatory; people-response

3. Which of the following scatterplots does not show a line of best ﬁt? b a.

b.

c.

4. Describe each of the following correlation coefﬁcients using the terms strong, moderate, or weak and positive or negative. a. r = 0.17 weak positive b. r = –0.62 moderate negative c. r = –0.88 strong negative d. r = 0.33 weak positive e. r = 0.49 moderate positive f. r = –0.25 weak negative g. r = 0.91 strong positive 5. The graph shows supply and demand curves for the newest game controller for the VVV video game system. a. What is the equilibrium price? $11.49 q b. What will happen if the price is set at $7.99? Demand will exceed supply. 1,000 c. How many game controllers are demanded at a price of $7.99? 1,000 d. How many game controllers are supplied at 250 a price of $7.99? 250 e. What will happen if the price is set at $7.99 $11.49 $12.99? Supply will exceed demand, and suppliers

Supply

Demand p

will attempt to sell the excess product at a lower price.

Assessment

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ANSWERS 6d. demand = 2,500; when price is set at $25 per item, the demand for the item will be 2,500. 7a.

9b. The maximum proﬁt price is approximately $23.33. At that price, the maximum proﬁt is approximately $76,666.66.

6. The demand function for a certain product is q = –300p + 10,000. The ﬁxed expenses are $500,000 and the variable expenses are $2 per item produced. a. Express the expense function in terms of q. E = 2q + 500,000 b. Use substitution to express the expense function in terms of p. E = –600p + 520,000 c. If the price is set at $20, what quantity will be demanded? 4,000 d. If price is set at $25, ﬁnd the demand. Use these numbers in a complete sentence that explains what they mean. See margin. e. If q = 1,000 widgets, ﬁnd E, the cost (expense) of producing them. Use both numbers in a complete sentence to explain what they mean. E = 502,000; it costs $502,000 to produce 1,000 widgets. f. If the price is set at p = $15, how much will it cost to produce the correct number of widgets? Use these numbers in a complete sentence to explain what they mean. E = 529,000; it would cost $529,000 to produce widgets that sell for $15 each.

7. Let the expense function for a particular item be deﬁned as y = –1,950x + 53,000. Let the revenue function be deﬁned as y = –450x2 + 10,000x, where x represents price in each equation. Use this window. a. Graph the two functions. See margin. WINDOW b. Determine the x- and y-coordinates of Xmin=0 Xmax=30 the ﬁrst point where the two graphs Xscl=1 intersect. Round those values to the Ymin=0 Ymax=70000 nearest whole number. x = 6 and y = 42,026 Yscl=1 c. Explain the signiﬁcance of this point in Xres=1 the context of expense, revenue, and price. When the price is set at approximately $6, both the revenue and the expense will be approximately $42,026.

8. Express the revenue equation in terms of price given the demand function. a. q = –900p + 120,000 R = –900p 2 + 120,000p b. q = –88,000p + 234,000,000 R = –88,000p 2 + 234,000,000p 9. The expense function for a widget is E = –3,000p + 250,000. The revenue function is R = –600p2 +25,000p. a. Write the proﬁt equation in simpliﬁed form. P = –600p2 + 28,000p – 250,000 b. Use the axis of symmetry formula to determine the maximum proﬁt price and the maximum proﬁt. See margin. Use this situation to answer Exercises 10–25. A company is interested in producing and selling a new device called an eyePOD (eyewear personal optical device). The eyePOD is an MP3 and video player built into a pair of sunglasses. The user can listen to music from the small earphones and watch videos projected on the screen behind the glasses. 10. As the price of the eyePOD increases, what is expected to happen to the quantity demanded? decreases 11. The horizontal axis represents price, and the vertical axis represents quantity. Does the demand function have a positive or negative slope? Explain. The demand is decreasing; the slope is negative.

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12. The market research department conducted consumer surveys at college campuses and reported its results. In these ordered pairs, the ﬁrst number represents price, p, and the second number represents the quantity demanded, q. The points are listed as ( p, q).

12.

(300, 10,000), (325, 8,900), (350, 8,800), (375, 8,650), (400, 6,700), (425, 6,500), (450, 5,000), (475, 4,500), (500, 4,450), (525, 3,000) Make a scatterplot of the data. See margin. 13. What is the correlation coefﬁcient? Round it to the nearest hundredth. Is this line a good predictor? Explain. –0.98; because the absolute value of the correlation coefﬁcient is greater than 0.75, this is a good predictor.

24. a. 7,360; b. $389.41; c. $161.13; d. $617.69; e. $2,865,877.15; f. $1,263,930.49; g. $1,601,946.66

14. Write the regression equation. Remember that the demanded quantity, q, is the dependent variable. Round the slope and y-intercept to the nearest hundredth. q = –30.74p + 19,330 15. The accounting department has calculated that this could be the biggest product to hit the market in years. It anticipates the ﬁxed costs to be $160,000 and the variable cost to be $150 per eyePOD. Express expenses, E, as a function of q, the quantity produced. E = 150q + 160,000 16. Express the revenue, R, in terms of p and q. R = pq 17. Express the revenue, R, in terms of p. R= –30.74p 2 + 19,330p 18. Recall the transitive property of dependence. Express expenses, E, in terms of p. Round to the nearest hundredth. E = –4,611p + 3,059,500 19. Graph the expense and revenue functions. a. Determine an appropriate maximum horizontal-axis value. 700 b. Determine an appropriate maximum vertical-axis value. 3,100,000 c. Sketch the graphs of the expense and revenue functions. See additional answers.

20. Determine the coordinates of the maximum point on the revenue graph. Round to the nearest hundredth. (3,14.41, 3,038,784.16) 21. Determine the breakeven points. Round to the nearest hundredth. (161.13, 2,316,534.30) and (617.69, 211,315.67)

22. Express the proﬁt, P, in terms of p. P = –30.74p 2 + 23,941p – 3,059,500 23. Graph the proﬁt function. Determine the coordinates of the maximum point of the proﬁt graph. At what price, p, is proﬁt maximized? Round to the nearest cent. This will be the price at which one eyePOD will sell! (389.41, 1,601,946.66); the proﬁt is maximized at a price of $389.41. 24. Write the business summary statement by ﬁlling in the blanks. In summary, to start this business, __a.__ eyePODS should be manufactured. Each should be sold at $__b.__. The breakeven point is reached at a price of $__c.__ or $__d.__, but a proﬁt is made at any price between those prices. At the selling price, there is a revenue of $__e.__ and expenses of $__f.__, resulting in a proﬁt of $__g.__. See margin. 25. If shares of stock are sold with an initial value of $10 each, how many shares must be sold to get enough money to start the business? 26,394

Assessment

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Modeling a Business - cnusd

2 CHAPTER Modeling a Business Business, more than any other occupation, is a continual dealing with the future; it is a continual calculation, an i...

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Modeling a Business - cnusd

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