Gerver R.K., Sgroi R.J. Financial Algebra
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104 Chapter 2 Modeling a Business
Mathematically modeling a business situation helps you better under-
stand 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.
SOLUTION E depends on q, and q depends on p. To fi nd the value of
E at a particular price, p, use substitution or express the expense equa-
tion directly in terms of price. Both methods illustrate the transitive
property of dependence.
The fi rst method uses the price given to fi nd a value for q that can be
substituted into the equation for E.
Method 1
Use the equation for q. q = 80p + 100,000
Substitute 60 for p. q = 80(60) + 100,000
Simplify. q = 104,800
Use the equation for E. E = 50q + 80,000
Substitute 104,800 for q. E = 50(104,800) + 80,000
Simplify. E = 5,320,000
The second method substitutes the value of q in terms of p into the
expense equation, and then uses the price given to fi nd the value of E.
Method 2
Use the equation for E. E = 50q + 80,000
Substitute 80p + 100,000 for q. E = 50(80p + 100,000) + 80,000
Distribute. E = 4,000p + 5,000,000 + 80,000
Calculate. E = 4,000p + 5,080,000
Substitute 60 for p. E = 4,000(60) + 5,080,000
Simplify. E = 5,320,000
The expenses total $5,320,000.
CHECK
■
YOUR UNDERSTANDING
Determine the expense, E, for production of an item when the price,
p, is $42 given E= 50q + 80,000 and q = 80p + 100,000.
EXTEND
■
YOUR UNDERSTANDING
Suppose A = 20x + 30, x = 30y + 40, and y = 40z + 50. Describe
how the value of A depends on the value of z.
Skills and Strategies
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2-8 Mathematically Modeling a Business 105
EXAMPLE 2
A business model uses a summary analysis of the situa-
tion in terms of dependent variables. Examine the three
graphs of a business situation for the production of wid-
gets. The graphs depict numerical information that is
needed to complete the summary analysis.
Write the summary analysis in terms of the data pre-
sented 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 profi t is made at any price between those
prices.
At the selling price, there is revenue of $__?__ and
expenses of $__?__, resulting in a profi t of $__?__.
SOLUTION In summary, to start this business,
103,200 widgets should be manufactured. Each should
be sold for $40.
This is asking the price that will yield the maximum
profi t and the quantity demanded at that price.
The breakeven point is reached at a price of $14.83 or
$65.17, but a profi t is made at any price between those
prices. These amounts can be found at the intersection
points of the expense and revenue functions.
At the selling price, there is revenue of $800,000 and
expenses of $420,000, resulting in a profi t of $380,000.
Use the price that yields the maximum profi t, $40, in both
the expense and profi t functions.
150,000
200
price
demand
0
Demand
(40, 103,200)
900,000
100
price
amount
0
Expense
(36.57, $806,666.67)
Revenue
(40, $420,000)
(14.83, 520,664.46)
(65.17, $319,335.54)
(40, $800,000)
CHECK
■
YOUR UNDERSTANDING
Use the points labeled on the graphs to show that the
maximum profi t at the selling price is the difference
between the revenue and expense values at that price.
900,000
100
price
prot
0
Prot
(40, 380,000)
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1. How might those words apply to what has been outlined in this
lesson?
2. Use the following graph to answer the questions below. Let Y
1
= cost
function and Y
2
= revenue function.
a. Explain the signifi cance of point (D, B).
b. Explain the signifi cance of point (E, A).
c. Explain the signifi cance of point (F, C).
d. Explain the signifi cance of point (G, 0).
e. Explain the signifi cance of point (H, 0).
f. Where do you think the maximum profi t might occur?
3. Draw a graph of cost function Y
1
and revenue function Y
2
that meet
the following criteria.
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 pro-
vider, 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 dif-
ferent prices.
4. As the price increases, what is expected to happen to the quantity
demanded?
5. The horizontal axis represents price, and the vertical axis represents
quantity. Does the demand function have a positive or negative
slope? Explain.
All models are wrong. Some models are useful.
George Box, Statistician and Quality Control Pioneer
Applications
0
DE FG H
A
B
C
106 Chapter 2 Modeling a Business
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2-8 Mathematically Modeling a Business 107
6. The company decides to conduct a market research survey to deter-
mine 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?
7. Find the regression equation. Remember the quantity demanded, q,
is the dependent variable. Round the slope and y–intercept to the
nearest hundredth.
8. Is the linear regression line a good predictor? Explain.
9. Examine the data to see if there is any relationship between the price
and the quantity demanded. Determine the correlation coeffi cient
between price and demand, rounded to nearest hundredth. Explain
the signifi cance of the correlation coeffi cient.
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.
11. Express the revenue, R, in terms of p and q.
12. Express the revenue, R, in terms of p.
13. Use the transitive property of dependence to express expense, E, in
terms of p. Round to the nearest hundredth.
14. Graph the expense and revenue functions.
a. Determine an appropriate maximum horizontal-axis value.
b. Determine an appropriate maximum vertical-axis value.
c. Sketch the graphs of the expense and revenue functions.
15. Determine the coordinates of the maximum point on the revenue
graph. Round the coordinates to the nearest hundredth.
16. Determine the breakeven points. Round to the nearest hundredth.
17. Express the profi t, P, in terms of p.
18. Graph the profi t function. Determine the coordinates of the maxi-
mum point of the profi t graph. At what price, p, is profi t maximized?
Round to the nearest cent. This will be the price at which Toejack
will sell!
19. Write the business summary statement by fi lling 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 profi t 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 profi t of $__g.__.
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?
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108 Chapter 2 Modeling a Business
CHAPTER
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/fi nancialalgebra. Copy and paste it
into your article.
You Write the Story!!
Real Numbers
Assessment
2
Reality Check
0 0.5 1 1.5 2 2.5
Product Failure Rate Over Time
Time in Years
3 3.5 4 4.5
Failure Rate
1. Corporations pride themselves on the recognition of their logos. This
project involves creating a game. Use the library or the Internet to fi nd
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 fi nd 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 specifi c 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 inven-
tion and its failure.
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4. Interview a local businessperson. Ask for examples of fi xed and vari-
able 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 fi gures, features, pictures,
model names, etc. Prepare the fi ndings in a report.
6. On January 1, 1962, the Beatles, looking for their fi rst recording con-
tract, 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
fi nancially-related Beatles facts. Compare Decca’s decision to the
product failures discussed in this chapter. What are the similarities
and differences? Prepare the fi ndings 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 fi ndings, and if they differ, fi nd out where they differ.
How did the fi ndings 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 iden-
tify three companies that claim to have made a profi t over a period
of time. Research how they each defi ne profi t, and state the monetary
amounts that each is quoting as their profi t for the time period.
10. Many children set up lemonade stands in front of their homes when
they were younger. What would the fi xed 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 profi t over a one-week period.
Go to www.cengage.com/school/math/fi nancialalgebra where you will fi nd a link to
a website containing current issues about modeling a business.
Assessment 109
Dollars and Sense
Your Financial News Update
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110 Chapter 2 Modeling a Business
REVISITED
Really!
Really?
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 condi-
tion of a classic car. On this graph, Condition 2 is the best condition and
Condition 6 is the poorest condition.
1. Give an approximate value for the 1958 Villager in a condition of
6 in 2008.
2. Give an approximate value for the 1958 Villager in a condition of
2 in 2008.
3. Approximate the difference between a 1958 Villager in a condition of
6 and in a condition of 2 in 2008.
4. Approximate the difference between a 1958 Villager in a condition of
2 in 1999 and in 2008.
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 pic-
tures of the original 1958 Edsel. Also fi nd 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!
234
16000
14000
12000
10000
8000
6000
4000
2000
0
5
Condition
1958 Villager 9-Passenger Wagon
Price (US$)
6
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
Source: Edsel.com, http://www.edsel.com/charts/e58vil9p.jpg
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1. Examine each scatterplot. Identify each as showing a positive corre-
lation, a negative correlation, or no correlation.
a.
b.
c.
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
b. calories burned and number of minutes of exercising
c. amount paid as income tax and the amount of a paycheck
d. pounds of hamburger used to make a meatloaf and number of
people that can be fed from the meatloaf
3. Which of the following scatterplots does not show a line of best fi t?
a.
b.
c.
4. Describe each of the following correlation coeffi cients using the
terms strong, moderate, or weak and positive or negative.
a. r = 0.17
b. r = –0.62
c. r = –0.88
d. r = 0.33
e. r = 0.49
f. r = –0.25
g. r = 0.91
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?
b. What will happen if the price is set
at $7.99?
c. How many game controllers are demanded
at a price of $7.99?
d. How many game controllers are supplied at
a price of $7.99?
e. What will happen if the price is set at
$12.99?
q
p
$11.49
Supply
Demand
$7.99
1,000
250
Applications
Assessment 111
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112 Chapter 2 Modeling a Business
6. The demand function for a certain product is q = –300p + 10,000.
The fi xed expenses are $500,000 and the variable expenses are $2 per
item produced.
a. Express the expense function in terms of q.
b. Use substitution to express the expense function in
terms of p.
c. If the price is set at $20, what quantity will be demanded?
d. If price is set at $25, fi nd the demand. Use these numbers in a
complete sentence that explains what they mean.
e. If q = 1,000 widgets, fi nd E, the cost (expense) of producing
them. Use both numbers in a complete sentence to explain what
they mean.
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.
7. Let the expense function for a particular item be defi ned as
y = –1,950x + 53,000. Let the revenue function be defi ned as
y = –450x
2
+ 10,000x, where x represents price in each equation.
Use this window.
a. Graph the two functions.
b. Determine the x- and y-coordinates of
the fi rst point where the two graphs
intersect. Round those values to the
nearest whole number.
c. Explain the signifi cance of this point in
the context of expense, revenue, and
price.
8. Express the revenue equation in terms of price given the demand
function.
a. q = –900p + 120,000
b. q = –88,000p + 234,000,000
9. The expense function for a widget is E = –3,000p + 250,000. The
revenue function is R = –600p
2
+25,000p.
a. Write the profi t equation in simplifi ed form.
b. Use the axis of symmetry formula to determine the maximum
profi t price and the maximum profi t.
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?
11. The horizontal axis represents price, and the vertical axis represents
quantity. Does the demand function have a positive or negative
slope? Explain.
Xmin=0
Xmax=30
WINDOW
Xscl=1
Ymin=0
Ymax=70000
Yscl=1
Xres=1
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12. The market research department conducted consumer surveys at
college campuses and reported its results. In these ordered pairs, the
fi rst number represents price, p, and the second number represents
the quantity demanded, q. The points are listed as ( p, q).
(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.
13. What is the correlation coeffi cient? Round it to the nearest hun-
dredth. Is this line a good predictor? Explain.
14. Write the regression equation. Remember that the demanded quan-
tity, q, is the dependent variable. Round the slope and y-intercept to
the nearest hundredth.
15. The accounting department has calculated that this could be the big-
gest product to hit the market in years. It anticipates the fi xed 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.
16. Express the revenue, R, in terms of p and q.
17. Express the revenue, R, in terms of p.
18. Recall the transitive property of dependence. Express expenses, E, in
terms of p. Round to the nearest hundredth.
19. Graph the expense and revenue functions.
a. Determine an appropriate maximum horizontal-axis value.
b. Determine an appropriate maximum vertical-axis value.
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.
21. Determine the breakeven points. Round to the nearest hundredth.
22. Express the profi t, P, in terms of p.
23. Graph the profi t function. Determine the coordinates of the maxi-
mum point of the profi t graph. At what price, p, is profi t maximized?
Round to the nearest cent. This will be the price at which one eye-
POD will sell!
24. Write the business summary statement by fi lling in the blanks.
In summary, to start this business, __a.__ eyePODS should be manu-
factured. Each should be sold at $__b.__. The breakeven point is
reached at a price of $__c.__ or $__d.
__, but a profi t 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 profi t of $__g.__.
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?
Assessment 113
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