Chapter 5.8, Problem 35PPSE

Solution

(a)

To calculate : The average rate of change between $1955\text{ and }1965$ and between $1975\text{ and }1985$ .

The average rate of change between $1955\text{ and }1965$ is and between $1975\text{ and }1985$ is .

Given information :

The table given below shows the percentage of U.S. labor force in unions for selected years between $1955\text{ and }2005$ .

Year	%
$\begin{array}{l}1955\\ 1960\\ 1965\\ 1970\\ 1975\\ 1980\\ 1985\\ 1990\\ 1995\\ 2000\\ 2005\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

Calculation :

The average rate of change between $1955\text{ and }1965$ is:

  $\begin{array}{l}=\frac{33.2-28.4}{28.4}\times 100\\ =16.9\%\end{array}$

The average rate of change between $1975\text{ and }1985$ is:

  $\begin{array}{l}=\frac{18-35.5}{35.5}\times 100\\ =-49.3\%\end{array}$

(b)

To determine : The type of polynomial model that seems to be most appropriate by making a scatter plot.

The cubic polynomial model seems to be most appropriate.

Given information :

The table given below shows the percentage of U.S. labor force in unions for selected years between $1955\text{ and }2005$ .

Year	%
$\begin{array}{l}1955\\ 1960\\ 1965\\ 1970\\ 1975\\ 1980\\ 1985\\ 1990\\ 1995\\ 2000\\ 2005\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

Explanation :

Let $x$ represent the years starting from $1955$ and $y$ represent the percentage of U.S. labor force in unions. The table will be

$x$	$y$
$\begin{array}{l}0\\ 5\\ 10\\ 15\\ 20\\ 25\\ 30\\ 35\\ 40\\ 45\\ 50\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

  

Therefore, the model will not be linear but it will be cubic.

(c)

To determine : The type of polynomial model using graphing calculator.

The required polynomial model is $y=2.27x^3\times {10}^{-4}-0.015x^2-0.217x+32.91$

Given information :

The table given below shows the percentage of U.S. labor force in unions for selected years between $1955\text{ and }2005$ .

Year	%
$\begin{array}{l}1955\\ 1960\\ 1965\\ 1970\\ 1975\\ 1980\\ 1985\\ 1990\\ 1995\\ 2000\\ 2005\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

Explanation :

The cubic model will be in the form of: $y=a{x}^3+b{x}^2+cx+d$ .

Use calculator to find the values of $a,b,c,\text{ and }d$ .

  $a=2.27\times {10}^{-4},b=-0.015,c=-0.217,\text{ and }d=32.91$ and the value of $r^2$ is $0.992$ . As the value of $r^2$ is very close to one, so the cubic model that best fits the given data is:

  $y=2.27x^3\times {10}^{-4}-0.015x^2-0.217x+32.91$ .

Therefore, the required function is $y=2.27x^3\times {10}^{-4}-0.015x^2-0.217x+32.91$ .

(d)

To calculate : The percent of the labor force in unions in the year 2020.

The percent of the labor force in unions in the year 2020 is $17.76$ .

Given information :

The table given below shows the percentage of U.S. labor force in unions for selected years between $1955\text{ and }2005$ .

Year	%
$\begin{array}{l}1955\\ 1960\\ 1965\\ 1970\\ 1975\\ 1980\\ 1985\\ 1990\\ 1995\\ 2000\\ 2005\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

Calculation :

The cubic model that best fits the given data is $y=2.27x^3\times {10}^{-4}-0.015x^2-0.217x+32.91$ .

To estimate percent of the labor force in unions in the year 2020, take $x=2020-1955=65$ .

So,

  $\begin{array}{c}y=2.27{\left(65\right)}^3\times {10}^{-4}-0.015{\left(65\right)}^2-0.217\left(65\right)+32.91\\ =62.34-63.38-14.11+32.91\\ =17.76\end{array}$

Therefore, the percent of the labor force in unions in the year 2020 is $17.76$ .

(e)

To explain : The confidence in the prediction.

The prediction is not very reliable.

Given information :

The table given below shows the percentage of U.S. labor force in unions for selected years between $1955\text{ and }2005$ .

Year	%
$\begin{array}{l}1955\\ 1960\\ 1965\\ 1970\\ 1975\\ 1980\\ 1985\\ 1990\\ 1995\\ 2000\\ 2005\end{array}$	$\begin{array}{l}33.2\\ 31.4\\ 28.4\\ 27.3\\ 35.5\\ 21.9\\ 18.0\\ 16.1\\ 14.9\\ 13.5\\ 12.5\end{array}$

Explanation :

The prediction made on the basis of above data is based on extrapolation.

Therefore, the prediction is not very reliable.