Chapter 1.7, Problem 62E

a.

To determine

Graph the function over the specified domain.

Answer to Problem 62E

  

Explanation of Solution

Calculation:

Find the average rate of change of the function from $2000to2014$ and predict the number of households in the United states in $2022$ .

The number $N$ of households model is $N(x)=-0.023{(x-33.12)}^2+131,0\le t\le 14$ where $t$ represents the year with $t=0$ corresponding to $2000$ .

The parent function $f$ is quadratic function $f(x)=x^2$ . The graph of $N$ is a reflection in the $x-axis$ followed by right shift of $33.12$ units of the graph of $f(x)=x^2$ and vertically compressed by a factor of $0.023$ , then the resultant graph is shifted up $131$ units. Using the graphing utility, the graph:

  

b.

To determine

Find the average rate of change of the function.

Answer to Problem 62E

  $\$1.1885million$

Explanation of Solution

Calculation:

Find the average rate of change of the function from $2000to2014$ and predict the number of households in the United states in $2022$ .

The number $N$ of households model is $N(x)=-0.023{(x-33.12)}^2+131,0\le t\le 14$ where $t$ represents the year with $t=0$ corresponding to $2000$ .

The average rate of change of the model from $2000to2014$ is given by

  $\left[\begin{array}{l}\text{The average}\\ \text{ rate of change}\end{array}\right]=\frac{f(14)-f(0)}{14-0}$

  $\begin{array}{l}=\frac{\left[-0.023{(14-33.12)}^2+131\right]-[-0.023{(0-33.12)}^2+131}{14}\\ =\frac{122.592-105.953}{14}\\ =1.1885\end{array}$

Hence, the average rate of change is $1.1885$ million.

c.

To determine

Use the model to predict the number of households.

Answer to Problem 62E

  $128.156$

Explanation of Solution

Calculation:

Find the average rate of change of the function from $2000to2014$ and predict the number of households in the United states in $2022$ .

The number $N$ of households model is $N(x)=-0.023{(x-33.12)}^2+131,0\le t\le 14$ where $t$ represents the year with $t=0$ corresponding to $2000$ .

Substitute $x=22$ into the given model

  $\begin{array}{l}$ N(x)=-0.023{(x-33.12)}^2+131$$ N(22)=-0.023{(22-33.12)}^2+131.$N(22)=128.156\end{array}$

Hence, $N(22)=128.156$ . This is less than the number of households from $2014$ .