Chapter 12.4, Problem 44E

a.

To determine

Graph the model.

Answer to Problem 44E

  

Explanation of Solution

Given information:

The table shows the numbers $N$ (in thousands) of U.S. military reserve personnel for the years 2000 through 2010.

A model for the data is given by

Y13y $N=\frac{1280.4-8818t}{1.0-0.04t-0.002t^2}$ ,

  $0\le t\le 10$

Where $t$ represents the year, with $t=10$ corresponding to 2000.

Use a graphing utility to create a scatter plot of the data and graph the model in the same viewing window. How well does the model fit the data?

Calculation:

Using $t$ as number of years since 2000, the given table can be replaced as

  $\begin{array}{l}\\ \begin{array}{cc}\begin{array}{l}\text{Years since 2}000\\ (t)\end{array}& \begin{array}{l}\text{Number of military reserve personnel(in thouasands)}\\ \left(N\right)\end{array}\\ 0& 1277\\ 1& 1249\\ 2& 1222\\ 3& 1189\\ 4& 1167\\ 5& 1136\\ 6& 1120\\ 7& 1110\\ 8& 1100\\ 9& 1094\\ 10& 1103\end{array}\end{array}$

Enter the data from given table into your graphing utility calculator.

To enter the data first press $STAT$ the key and then press $1$ to select $Edit$ .

  

  

Now input all the data given sets followed by $ENTER$ key.

  

  

Before you use regression feature, make sure to enter the DignosticOn command stored in $CATALOG$ menu.

The calculator does not show the value of $r$ regression coefficient unless the DignosticOn command has been entered.

  

  

To plot the data, press the $STAT$

  $PLOT$ key,

  

Press $ENTER$ key to select $PLOT$

  $1$ . Your cursor should be flashing on the word On.

Now press $ENTER$ key.

  

Now press $ZOOM$ key and select option $9$ : ZoomStat to have the calculator set the new scale for the axes.

  

The plot of the data will appear as below:

  

To graph the model $N=\frac{1280.4-8818t}{1.0-0.04t-0.002t^2}$ in the same viewing window, press the $Y=$ and the input the function $Y_1=\left(1280.4-88.18X\right)/\left(1.0-0.04X-0.002X^2\right)$

  

Press the $GRAPH$ key to graph the model in the same viewing window of the scatter plot.

  

The graph of the model does not pass through all the points, some points at the end are far apart from the model curve.

Hence, the model does not fit the data after year 2000

b.

To determine

Predict the number of military reserve personnel in 2017.

Answer to Problem 44E

  $\boxed{848}$ thousands.

Explanation of Solution

Given information:

The table shows the numbers $N$ (in thousands) of U.S. military reserve personnel for the years 2000 through 2010.

A model for the data is given by

  $N=\frac{1280.4-8818t}{1.0-0.04t-0.002t^2}$ , $0\le t\le 10$

Where $t$ represents the year, with $t=10$ corresponding to 2000.

Use the model to predict the number of military reserve personnel in 2017.

Calculation:

To find the average monthly benefit in 2017, replace $t$ by $17$ in the function $N=\frac{1280.4-8818t}{1.0-0.04t-0.002t^2}$ and then evaluate.

  $N=\frac{1280.4-8818\left(17\right)}{1.0-0.04\left(17\right)-0.002{\left(17\right)}^2}$

  $\begin{array}{l}=\frac{-218.66}{-0.258}\\ \approx 848\end{array}$

Hence, the average monthly benefit in 2017 is about $\boxed{848}$ thousands.

c.

To determine

What is the limit of the function as approaches infinity?

Do you think the limit is realistic?

Answer to Problem 44E

  $\boxed{0}$

  $not$ realistic.

Explanation of Solution

Given information:

The table shows the numbers $N$ (in thousands) of U.S. military reserve personnel for the years 2000 through 2010.

A model for the data is given by

  $N=\frac{1280.4-8818t}{1.0-0.04t-0.002t^2}$ , $0\le t\le 10$

Where $t$ represents the year, with $t=10$ corresponding to 2000.

What is the limit of the function as approaches infinity? Explain the meaning of the limit in the context of the problem. Do you think the limit is realistic? Explain

Calculation:

Determine the limit of average monthly benefit function as $t$ approaches infinity, that is, find the limit

  $\underset{t\to \infty }{\lim }\frac{1280.4-88.18t}{1.0-0.04t+0.002t^2}$

  $=\underset{t\to \infty }{\lim }\frac{\frac{1280.4}{t^2}-\frac{88.18}{t}}{\frac{1.0}{t^2}-\frac{0.04}{t}+0.002}$

  $\begin{array}{l}=\frac{0-0}{0-0-0.002}\\ =0\end{array}$

Hence, the limit of military reserve personnel function as $t$ approaches infinity is $\boxed{0}$ which means as the time increases, the number of military reserve personnel approaches to $\boxed{0}$ .

As time increases, , the number of military reserve personnel will decrease and then again increase and it will never become $0$ .

Hence, the model is $not$ realistic.