Chapter 1.4, Problem 72E

(a)

To determine

The value of $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$

Answer to Problem 72E

  $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}=6.54\text{millions/year}$

Explanation of Solution

Given information:

Given that the value is $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$ , interpret the result in the context of the problem and make a scatter plot of the problem.

Formula used:

Subtraction followed by division is performed.

Calculation:

Use the table to find $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$

  $\begin{array}{l}\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}=\frac{125.8-80}{7}\\ =\frac{45.8}{7}\\ \frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}=6.54\text{millions/year}\end{array}$

A logical interpretation is increase in number of tax return filling per year.

Conclusion:

A logical interpretation is increase in number of tax return filling per year.

(b)

To determine

The scatter the plot

Answer to Problem 72E

The value gets increased continuously.

Explanation of Solution

Given information:

Given that the value is $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$ , interpret the result in the context of the problem and make a scatter plot of the problem.

Formula used:

The x axis represents the number of years and the y axis represent the number of tax returns.

Calculation:

Make a scatter plot.

  

Conclusion:

The value gets increased continuously.

(c)

To determine

The linear model for the data algebraically.

Answer to Problem 72E

  $N=6.54t+34.22$

Explanation of Solution

Given information:

Given that the value is $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$ , interpret the result in the context of the problem and make a scatter plot of the problem.

Formula used:

  $N-N_1=\frac{N_2-N_1}{t_2-t_1}\left(t-t_1\right)$

Calculation:

Let take two points $\left(N_1,t_1\right)=\left(7,80\right)\text{and }\left(N_2,t_2\right)=\left(14,125.8\right)$ . Use the point-slope formula and have

  $\begin{array}{l}N-N_1=\frac{N_2-N_1}{t_2-t_1}\left(t-t_1\right)\\ N-80=\frac{125.8-80}{14-7}\left(t-7\right)\\ N-80=6.54\left(t-7\right)\\ N=6.54t+34.22\end{array}$

Conclusion:

The linear model of the data is $N=6.54t+34.22$

(d)

To determine

The comparison of value with the actual value.

Answer to Problem 72E

The value get increasing continuously on $N=6.54t+34.22$

Explanation of Solution

Given information:

Given that the value is $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$ , interpret the result in the context of the problem and make a scatter plot of the problem.

Formula used:

  $N-N_1=\frac{N_2-N_1}{t_2-t_1}\left(t-t_1\right)$

Calculation:

Use the model found part (c) and the table will be

  

Conclusion:

The value get increasing continuously on $N=6.54t+34.22$

(e)

To determine

The linear model of the data.

Answer to Problem 72E

The linear model is $N\left(t\right)=6.54t+34.22$

Explanation of Solution

Given information:

Given that the value is $\frac{f\left(2014\right)-f\left(2007\right)}{2014-2007}$ , interpret the result in the context of the problem and make a scatter plot of the problem.

Formula used:

  $N-N_1=\frac{N_2-N_1}{t_2-t_1}\left(t-t_1\right)$

Calculation:

The values from the model are approximately the same as the actual data.

(f) Using graph utility, the linear model is $N\left(t\right)=6.54t+34.22$ the model derived from (c) is almost the same as the model given by the graphing utility.

Conclusion:

The linear model is $N\left(t\right)=6.54t+34.22$