Chapter 4, Problem 31RE

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

Chapter
Section

### Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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# Profit The net profits P (in millions of dollars) of Express Scripts from 2004 through 2013 are shown in the table. Year 2004 2005 2006 2007 2008 Profit 300.1 388.1 468.7 620.6 779.6 Year 2009 2010 2011 2012 2013 Profit 826.5 1204.6 1275.8 1340.5 1898.2 Spreadsheet at Larson Applied Calculus.comA model for these data is given by P =   147.8 e 0. l 96 t , where t represents the year, with t = 4 corresponding to 2004. (Source: Express Scripts Holding Co.)(a) How well does the model fit the data?(b) Find a linear model for the data. How well does the linear model fit the data? Which model, exponential or linear, is a better fit?(c) Use both models to predict the net profit in 2018.

(a)

To determine

Whether the model will be fitted in the data for the period 20042013 when the table representing the net profits P (in millions of dollars) earned by Express Scripts during the period 20042013 is as follows:

 Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Profit 300.1 388.1 468.7 620.6 779.6 826.5 1204.6 1275.8 1340.5 1898.2

The net profit P is given by the model as,

P=147.8e0.196t

Where, t represents the year in which t=4 corresponds to 2004.

Explanation

Given information:

The table representing the net profits P (in millions of dollars) earned by Express Scripts during the period 20042013 is as follows:

 Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Profit 300.1 388.1 468.7 620.6 779.6 826.5 1204.6 1275.8 1340.5 1898.2

The net profit P is given by the model as,

P=147.8e0.196t

Where, t represents the year in which t=4 corresponds to 2004.

Consider the model, P=147.8e0.196t

Where, t represents the year in which t=4 corresponds to 2004.

Compute the profit in 2004 by substituting t=4 in the model as,

P(4)=147.8e0.196(4)=147.8e0.784147.82.190216$323.7 million Compute the profit in 2005 by substituting t=5 in the model as, P(5)=147.8e0.196(5)=147.8e0.98147.82.664456$393.8 million

Compute the profit in 2006 by substituting t=6 in the model as,

P(6)=147.8e0.196(6)=147.8e1.176147.83.241383$479.1 million Compute the profit in 2007 by substituting t=7 in the model as, P(7)=147.8e0.196(7)=147.8e1.372147.83.94323$582

(b)

To determine

To calculate: The linear model for the provided data and determine whether the linear model will be fitted in the data for the period 20042013 and also determine the best fit model from the given exponential model and the obtained linear model when the table representing the net profits P (in millions of dollars) earned by Express Scripts during the period 20042013 is as follows:

 Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Profit 300.1 388.1 468.7 620.6 779.6 826.5 1204.6 1275.8 1340.5 1898.2

The net profit P is given by the model as,

P=147.8e0.196t

Where, t represents the year in which t=4 corresponds to 2004.

(c)

To determine

To calculate: The net profit of the company in the year 2018 by using the model obtained in part (b) and the provided model when the table representing the net profits P (in millions of dollars) earned by Express Scripts during the period 20042013 is as follows:

 Year 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 Profit 300.1 388.1 468.7 620.6 779.6 826.5 1204.6 1275.8 1340.5 1898.2

The net profit P is given by the model as,

P=147.8e0.196t

Where, t represents the year in which t=4 corresponds to 2004.

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