Chapter 12.4, Problem 44E

a.

To determine

To create: A scatter plot of the data and graph the model in the same viewing window.

Answer to Problem 44E

The model is a good fit for the actual data values especially on the last half part.

Explanation of Solution

Given information: The table shows the numbers N(in thousands) of U.S. military.

receiving retired pay for the years 2000 through 2012

  

A model for the data is

  $N=\frac{1680.4-54.17t}{1.0-0.047t+0.00067t^2},0\le t\le 12.$

Where t represents the year with t=0 corresponding to 2000.

Calculation:

  

The model is a good fit for the actual data values especially on the last half part.

b.

To determine

To predict: The number of military retirees in 2019, using model.

Answer to Problem 44E

  $N\approx 1867\to 1867000retirees$ .

Explanation of Solution

Given information: The table shows the numbers N(in thousands) of U.S. military.

receiving retired pay for the years 2000 through 2012

  

A model for the data is

  $N=\frac{1680.4-54.17t}{1.0-0.047t+0.00067t^2},0\le t\le 12.$

Where t represents the year with t=0 corresponding to 2000.

Calculation:

  $\begin{array}{l}N=\frac{1680.4-54.17t}{1.0-0.047t+0.00067t^2},0\le t\le 12.\\ Att=19\to 2019.\\ N=\frac{1680.4-54.17(19)}{1.0-0.047t(19)+0.00067{(19)}^2}\\ N\approx 1867\to 1867000retirees\end{array}$

c.

To determine

The limit of the function as t approaches infinity and the limit in the context of the problem.

Answer to Problem 44E

  $\underset{\delta t\to \infty }{\lim }N=0.$

Explanation of Solution

Given information: The table shows the numbers N(in thousands) of U.S. military.

receiving retired pay for the years 2000 through 2012

  

A model for the data is

  $N=\frac{1680.4-54.17t}{1.0-0.047t+0.00067t^2},0\le t\le 12.$

Where t represents the year with t=0 corresponding to 2000.

Calculation:

  $\begin{array}{l}\underset{\delta t\to \infty }{\lim }N=\underset{\delta t\to \infty }{\lim }\frac{1680.4-54.17t}{1.0-0.047t+0.00067t^2}.\\ \underset{\delta t\to \infty }{\lim }N=\underset{\delta t\to \infty }{\lim }\frac{\frac{1680.4}{t^2}-\frac{54.17}{t}}{\frac{1.0}{t^2}-\frac{0.047}{t}+0.00067}\\ \underset{\delta t\to \infty }{\lim }N=\underset{\delta t\to \infty }{\lim }\frac{0-0}{0-0+0.0067}.\\ \underset{\delta t\to \infty }{\lim }N=0.\end{array}$

Over time ,the number of U.S. military retirees approach 0.This limit is not realistic given that it is necessary to have military in every country.