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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919

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Calculus: An Applied Approach (Min...

10th Edition
Ron Larson
ISBN: 9781305860919
Textbook Problem
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Modeling Data Personal consumption expenditures (in billions of dollars) for several types of recreation from 2008 through 2013 are shown in the table, where a is the expenditures on amusement parks and campgrounds, y is the expenditures on live entertainment (excluding sports), and z is the expenditures on spectator sports.

Year 2008 2009 2010 2011 2012 2013
X 38.3 37.2 38.8 41.3 44.4 46.5
y 24.3 25.2 26.3 28.3 28.4 28.7
z 19.4 18.8 19.2 20.4 21.2 22.3

Spreadsheet at LarsonAppliedCalculus.com

A model for the data in the table is given by

0.391 x + 0.069 x +   z   = 6.

(Source: U.S. Bureau of Economic Analysis)

(a) Add a fifth row to the table using the model to approximate z for the given values of x and y. Compare the approximations with the actual values of z.

(b) According to this model, increases in expenditures of recreation types y and z would correspond to what kind of change in expenditures of recreation type x?

(a)

To determine

To calculate: The fifth row with the help of model to approximate z for the provided values of x and y and then compare the approximate value to the actual values of z.

Explanation

Given information:

There are personal consumption expenditure from 2008 to 2013in which many type of recreation are shown in the table.

The provided table is

Year 2008 2009 2010 2011 2012 2013
x 38.3 37.2 38.8 41.3 44.4 46.5
y 24.3 25.2 26.3 28.3 28.4 28.7
z 19.4 18.8 19.2 20.4 21.2 22.3

The provided model equation for the data is 0.391x+0.069y+z=6.

Here, x is the expenditures on amusement park and campgrounds, y is the expenditures on live entertainment and z is the expenditures on spectators sports.

Calculation:

Consider the provided equation, 0.391x+0.069y+z=6

The approximate value of z can be found out by putting the values of x and y which is provided in the table in the above equation.

For 2008, put the value of x=38.3 and y=24.3 in the above equation.

0.391x+0.069y+z=60.391(38.3)+0.069(24.3)+z=614.9753+1.6767+z=6z=6+14.97531.6767

On further simplification,

z=6+14.97531.6767z=19.2986z19.3

For 2009, put the value of x=37.2 and y=25.2 in the above equation.

0.391x+0.069y+z=60.391(37.2)+0.069(25.2)+z=614.5452+1.7388+z=6z=6+14.54521.7388

On further simplification,

z=6+14.54521.7388z=18.8064z18.8

For 2010, put the value of x=38.8 and y=26.3 in the above equation.

0.391x+0.069y+z=60.391(38.8)+0.069(26.3)+z=615.1708+1.8147+z=6z=6+15.17021.8147

On further simplification,

z=6+15

(b)

To determine

The change in expenditures of recreation type x if increases in expenditures of recreation type y and z.

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