Copyright © Cengage Learning. All rights reserved.
SECTION
7.5
Choosing a
Mathematical Model
Copyright © Cengage Learning. All rights reserved.
Learning Objectives
1 Select the “best” function to model a real-world
data set given in equation, graph, table, or word
form
2 Use the appropriate function to model real-world
data sets
3 Use appropriate models to predict and interpret
unknown results
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Choosing a Mathematical Model
from a Table of Values
4
Choosing a Mathematical Model from a Table of Values
A mathematical model is a graphical, verbal, numerical,
and symbolic representation of a problem situation.
The model helps us understand the nature of the problem
situation and make predictions.
Mathematical models are used frequently to represent
real-world situations so it is important to master the process
of choosing an appropriate model.
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Example 1 – Choosing a Model from a Table of Values
The Fair Labor Standards Act was passed by Congress in
1938 to set a minimum hourly wage for American workers.
Table 7.23 displays the wages set for the year 1938 and
each year in which the wage was raised.
Table 7.23
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Example 1 – Choosing a Model from a Table of Values
cont’d
a. Determine whether a linear, a quadratic, or an
exponential function is the most appropriate
mathematical model for these data. Explain as you go
along why you did not choose the two types of functions
you reject.
b. Use your model to predict the minimum wage in 2008.
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Example 1(a) – Solution
To determine if these data are best
modeled by a linear function or a
nonlinear function (quadratic or
exponential), we first calculate the
average rate of change over each
time interval, as shown in
Table 7.24. (Note that the
time intervals are not equal.)
Table 7.24
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Example 1(a) – Solution
cont’d
We know that to be effectively modeled with a linear
function, the data must have a relatively constant rate of
change.
This data set has neither a constant nor nearly constant
rate of change so we rule out the linear model. To confirm
our conclusion, we draw the scatter plot in Figure 7.31.
U.S. Minimum Wage
Figure 7.31
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Example 1(a) – Solution
cont’d
The overall trend seems to show the data increasing at
an increasing rate and, therefore, it is nonlinear.
We next check to see if the data may be best modeled
by a quadratic function.
Using quadratic regression, we
find the quadratic model of best
fit,
and graph it in Figure 7.32.
U.S. Minimum Wage
Figure 7.32
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Example 1(a) – Solution
cont’d
The graph of the quadratic function appears to be a good
fit, but exponential functions can also model data that
are increasing at an increasing rate.
To test the exponential model,
we calculate the annual growth
factors to see if they are relatively
constant, as shown in Table 7.25.
Table 7.25
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Example 1(a) – Solution
cont’d
Table 7.25 (continued)
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Example 1(a) – Solution
cont’d
As we know that, to find the annual growth factor over a
time span greater than 1 year, we must find the nth root of
the quotient between two consecutive years.
For example, for the growth over the years from 1939 and
1945, we find the 6-year growth factor of
and
the annual growth factor of
Since the growth factors are nearly constant—ranging
between 1.01 and 1.20—we use exponential regression
to determine the exponential model of best fit:
.
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Example 1(a) – Solution
cont’d
Its graph is shown (in red) in Figure 7.33, along with the
quadratic model (in blue).
U.S. Minimum Wage
Figure 7.33
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Example 1(a) – Solution
cont’d
Like the quadratic model, the exponential model appears to
fit the trend in the data relatively well over the domain from
1938 to 1998.
To determine which of the two functions is the best choice,
we need to consider which give us the most reasonable
estimate for the minimum wage in years after 1998.
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Example 1(a) – Solution
cont’d
To find out, we expand the graphs to year 75 (or 2013) as
shown in Figure 7.34.
U.S. Minimum Wage
Figure 7.34
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Example 1(a) – Solution
cont’d
Now we can see that the exponential function departs from
the trend in the data quite markedly while the quadratic
function continues to model the trend in the data
reasonably well.
Therefore, we choose the quadratic function,
as the most appropriate model.
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Example 1(b) – Solution
cont’d
To predict the minimum wage in year 70 (or 2008), we use
the model from part (a) to evaluate Q(70).
We estimate the minimum wage in 2008 (year 70) will be
$6.87.
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Choosing a Mathematical Model from a Table of Values
The following strategies are useful when you are given a
table of data to model.
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Choosing a Mathematical Model from a Table of Values
In accomplishing Step 3, it is helpful to recognize key
graphical features exhibited by the data set, especially the
concavity of the scatter plot. Table 7.26 summarizes these
features.
Table 7.26
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Example 3 – Choosing a Model from a Table of Values
As shown in Table 7.28, the average price of a movie ticket
increased between 1975 and 2005. Find a mathematical
model for the data and forecast the average price of a
movie ticket in 2010.
Table 7.28
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Example 3 – Solution
We first draw the scatter plot shown in Figure 7.36.
Figure 7.36
The first four data points appear to be somewhat linear so
our initial impression is that a linear model may fit the data
well.
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Example 3 – Solution
cont’d
However, the fifth data point is not aligned with the first
four, so we know the data set is not perfectly linear.
However, a linear model may still fit the data fairly well.
We could also conclude that the scatter plot is concave
down on the interval
and concave up on
.
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Example 3 – Solution
cont’d
Since the scatter plot appears to change concavity once
and does not have any horizontal asymptotes, a cubic
model may work well.
In 2010, t = 35. Evaluating each function at t = 35 yields
The linear model predicts a ticket price of $6.93, and the
cubic model predicts a price of $8.43.
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Example 3 – Solution
cont’d
It is difficult to know which of these estimates of the
average price of a movie ticket is the most accurate since
both models seem to fit the data equally well.
Additionally, we have no other information that would lead
us to believe one model would be better than the other.
Because both models seem to fit the data equally well, we
choose the simplest one, which is
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Choosing a Mathematical Model from a Table of Values
It is important to note that in this type of problem we are not
trying to “hit” each data point with our model. Rather, we
are attempting to capture the overall trend.
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Choosing a Mathematical Model
from a Verbal Description
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Choosing a Mathematical Model from a Verbal Description
Certain verbal descriptions often hint at a particular
mathematical model to use.
By watching for key phrases,
we can narrow the model
selection process.
Table 7.29 presents some
typical phrases, their
interpretation, and possible
models.
Table 7.29
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Example 4 – Choosing a Model from a Verbal Description
In its 2001 Annual Report, the Coca-Cola Company
reported:
“Our worldwide unit case volume increased 4 percent
in 2001, on top of a 4 percent increase in 2000. The
increase in unit case volume reflects consistent
performance across certain key operations despite
difficult global economic conditions. Our business
system sold 17.8 billion unit cases in 2001.”
Find a mathematical model for the unit case volume of the
Coca-Cola Company.
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Example 4 – Solution
Since the unit case volume is increasing at a constant
percentage rate (4%), an exponential model should fit the
data well.
We have
,
where t is the number of years since the end of 2001
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Example 4 – Solution
cont’d
The report says that the initial number of unit cases sold
was 17.8 billion, so
The mathematical model for the unit case volume is
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Choosing a Mathematical Model from a Verbal Description
Sometimes a data set cannot be effectively modeled by
any of the aforementioned functions.
In these cases, we look to see if we can model the data
with a piecewise function.
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