Chapter 11.4, Problem 59E

a.

To determine

To create: a scatter plot of data and graph the model in the same viewing window and how do they compare.

Answer to Problem 59E

The model fit the data well.

Explanation of Solution

Given information: A model for the data is given by:

  $\mathrm{F}(\mathrm{t})=\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2},0\le \mathrm{t}\le 6$.

Where t represents the year, with t = 0 corresponding to 2010.

Calculation:

The graph of the model is shown below.

  

The model fit the data well.

b.

To determine

To predict: the numbers of female selected reserves in 2017 and 2018.

Answer to Problem 59E

  $\mathrm{F}(7)=157,889\text{female}\text{Selected}\text{Reserves}\text{.}$

  $\mathrm{F}(8)=159,128\text{female}\text{Selected}\text{Reserves}\text{.}$

Explanation of Solution

Given information: $\mathrm{F}(\mathrm{t})=\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2},0\le \mathrm{t}\le 6$

Calculation:

  $\begin{array}{c}\mathrm{F}(\mathrm{t})=\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2}\\ 2017\Rightarrow \mathrm{t}=7\Rightarrow \mathrm{F}(7)=\frac{153.06-23.674\times 7}{1-0.1597\times 7-0.00077{(7)}^2}=157.889\text{thousand}\\ \mathrm{F}(7)=157,889\text{female}\text{Selected}\text{Reserves}\text{.}\\ 2018\Rightarrow \mathrm{t}=8\Rightarrow \mathrm{F}(8)=\frac{153.06-23.674\times 8}{1-0.1597\times 8-0.00077{(8)}^2}=159.128\text{thousand}\\ \mathrm{F}(8)=159,128\text{female}\text{Selected}\text{Reserves}\text{.}\end{array}$

c.

To determine

To find: the limit of the model as $\mathrm{t}\to \mathrm{\infty }$ and interpret its meaning in the context of the situation.

Answer to Problem 59E

As time passes, the number of United States military female Selected Reserve personal approaches 0.

Explanation of Solution

Given information: $\mathrm{F}(\mathrm{t})=\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2},0\le \mathrm{t}\le 6$

Calculation:

  $\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\mathrm{F}(\mathrm{t})=\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2}=0.$

As time passes, the number of United States military female Selected Reserve personal approaches 0.

d.

To determine

To find: is this a good model for predicting future numbers of female Selected Reserves, explain.

Answer to Problem 59E

No, this is not a good model for predicting future numbers of female Selected Reserves because $\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\mathrm{F}(\mathrm{t})=\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2}=0.$

Explanation of Solution

Given information:

Calculation:

No, this is not a good model for predicting future numbers of female Selected Reserves because $\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\mathrm{F}(\mathrm{t})=\underset{\mathrm{t}\to \mathrm{\infty }}{\lim }\frac{153.06-23.674\mathrm{t}}{1-0.1597\mathrm{t}-0.00077{\mathrm{t}}^2}=0.$