Modeling a Business

Transcription

Modeling a Business

2
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
49657_02_ch02_p062-113.indd 62
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.
5/29/10 3:16:52 PM
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
63
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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
64
Chapter 2
49657_02_ch02_p062-113.indd 64
Modeling a Business
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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.
2-1
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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.
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.
66
Chapter 2
49657_02_ch02_p062-113.indd 66
As the x-values increase, the y-values decrease. Therefore,
this scatterplot shows a negative correlation between the two variables.
SOLUTION
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?
Modeling a Business
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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.
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.
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?
2-1
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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.
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
Chapter 2
49657_02_ch02_p062-113.indd 68
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
Modeling a Business
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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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49657_02_ch02_p062-113.indd 69
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.
69
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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?
49657_02_ch02_p062-113.indd 70
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
5/29/10 3:17:32 PM
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.
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.
Approximately how many more water bottles will
Rachael sell if the temperature increases 2 degrees?
2-2
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Linear Regression
71
5/29/10 3:17:36 PM
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.
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.
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.
72
Chapter 2
49657_02_ch02_p062-113.indd 72
Modeling a Business
5/29/10 3:17:40 PM
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
2-2
49657_02_ch02_p062-113.indd 73
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
73
5/29/10 3:17:44 PM
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.
74
Chapter 2
49657_02_ch02_p062-113.indd 74
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
5/29/10 3:17:48 PM
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.
2-3
49657_02_ch02_p062-113.indd 75
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
75
5/29/10 3:17:49 PM
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
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.
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.
The wholesale price of an item is x dollars. The retail price is r dollars.
Express the markup algebraically.
76
Chapter 2
49657_02_ch02_p062-113.indd 76
Modeling a Business
5/29/10 3:17:53 PM
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.
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
Use the graph to explain what happens if the price is set at $15.00.
EXAMPLE 4
A company wants to base the price of its product on demand for
the product, as well as on expenses. It takes a poll of several of its
current retailers to find out how many widgets they would buy at
different wholesale prices. The results are shown in the table. The
company wants to use linear regression to create a demand function.
What is the equation of the demand function? Round the slope and
y-intercept to the nearest hundredth.
SOLUTION
Use the linear regression feature on your graphing calculator. The equation is q = –1,756.19p + 30,238.82. This represents
the demand function.
Explain why it makes sense that the demand function has a negative
slope.
2-3
49657_02_ch02_p062-113.indd 77
linear regression line is an
“average” line, and if they
substitute the wholesale
prices from the table into
the regression equation,
they will not get the quantity
shown in the table, unless
the regression line goes
through the point with those
coordinates. That is unlikely.
Wholesale
Price ($) (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
77
5/29/10 3:17:54 PM
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
product at a higher
price.
6c. Supply will exceed
demand, and suppliers
excess product at a
lower price.
78
Chapter 2
49657_02_ch02_p062-113.indd 78
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
5/29/10 3:17:55 PM
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
2-3
49657_02_ch02_p062-113.indd 79
Supply and Demand
79
5/29/10 3:17:59 PM
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.
80
Chapter 2
49657_02_ch02_p062-113.indd 80
Modeling a Business
5/29/10 3:18:00 PM
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
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.
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
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.
The expense function for a certain product is E = 3.40q + 189,000.
Express the average cost of producing q items algebraically.
2-4
49657_02_ch02_p062-113.indd 81
81
5/29/10 3:18:03 PM
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.
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
82
Chapter 2
49657_02_ch02_p062-113.indd 82
Modeling a Business
5/29/10 3:18:05 PM
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.
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)
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.
Find the breakeven point for the expense function, E = 5.00q + 60,000,
and the revenue function, R = 7.00q.
2-4
49657_02_ch02_p062-113.indd 83
83
5/29/10 3:18:06 PM
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
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
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
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
84
Chapter 2
49657_02_ch02_p062-113.indd 84
Modeling a Business
5/29/10 3:18:08 PM
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
2-4
49657_02_ch02_p062-113.indd 85
85
5/29/10 3:18:11 PM
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
5/29/10 3:18:13 PM
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
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
2-5
49657_02_ch02_p062-113.indd 87
TEACH
87
5/29/10 3:18:16 PM
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
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.
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
5/29/10 3:18:17 PM
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.
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?
2-5
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.
89
5/29/10 3:18:19 PM
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.
90
Chapter 2
49657_02_ch02_p062-113.indd Sec12:90
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
5/29/10 3:18:26 PM
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
2-6
49657_02_ch02_p062-113.indd Sec13:91
Breakeven Analysis
91
5/29/10 3:18:29 PM
CLASS DISCUSSION
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.
92
Chapter 2
49657_02_ch02_p062-113.indd Sec13:92
Modeling a Business
5/29/10 3:18:33 PM
________
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
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
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
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.
How could you have improved on the error when calculating the
expense and revenue values?
2-6
49657_02_ch02_p062-113.indd Sec13:93
Breakeven Analysis
93
5/29/10 3:18:34 PM
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
What cell formulas were used to identify the a, b, and c values shown
in B11, B12, and B13?
94
Chapter 2
49657_02_ch02_p062-113.indd Sec13:94
Modeling a Business
5/29/10 3:18:36 PM
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.
2-6
49657_02_ch02_p062-113.indd Sec13:95
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
95
5/29/10 3:18:37 PM
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
96
Chapter 2
49657_02_ch02_p062-113.indd Sec13:96
price
0
200
Modeling a Business
5/29/10 3:18:39 PM
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.
2-7
49657_02_ch02_p062-113.indd Sec13:97
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.
The Profit Equation
97
5/29/10 3:18:40 PM
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.
98
Chapter 2
49657_02_ch02_p062-113.indd Sec13:98
Modeling a Business
5/29/10 3:18:44 PM
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
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
Sketch the graph. Identify the points where zero profit is made.
Explain your reasoning.
2-7
49657_02_ch02_p062-113.indd Sec13:99
The Profit Equation
99
5/29/10 3:18:50 PM
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
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
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.
Use the profit function from Example 1 Check Your Understanding.
Determine the price to the nearest cent that yields the maximum profit.
100
Chapter 2
49657_02_ch02_p062-113.indd Sec13:100
Modeling a Business
5/29/10 3:18:51 PM
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
2-7
49657_02_ch02_p062-113.indd Sec13:101
The Profit Equation
101
5/29/10 3:18:53 PM
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.
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.
16. FlipFlops manufactures beach sandals. Their expense and revenue
functions are E = –300p + 32,000 and R = –275p2 + 6,500p.
102
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Modeling a Business
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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.
2-8
49657_02_ch02_p062-113.indd Sec13:103
• 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.
103
5/29/10 3:18:57 PM
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.
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
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
E = 4,000(60) + 5,080,000
Simplify.
E = 5,320,000
The expenses total $5,320,000.
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.
Suppose A = 20x + 30, x = 30y + 40, and y = 40z + 50. Describe
how the value of A depends on the value of z.
104
Chapter 2
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Modeling a Business
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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.
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.
2-8
49657_02_ch02_p062-113.indd Sec13:105
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
105
5/29/10 3:19:02 PM
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.
106
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Modeling a Business
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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.
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
49657_02_ch02_p062-113.indd Sec13:107
107
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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.
108
Chapter 2
49657_02_ch02_p062-113.indd Sec13:108
Modeling a Business
5/29/10 3:19:08 PM
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
49657_02_ch02_p062-113.indd Sec13:109
109
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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.
110
Chapter 2
49657_02_ch02_p062-113.indd Sec13:110
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
5/29/10 3:19:10 PM
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
49657_02_ch02_p062-113.indd Sec13:111
111
5/29/10 3:19:11 PM
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.
112
Chapter 2
49657_02_ch02_p062-113.indd Sec13:112
Modeling a Business
5/29/10 3:19:13 PM
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.
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
49657_02_ch02_p062-113.indd Sec13:113
113
5/29/10 3:19:14 PM

Modeling a Business

Transcription

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