Chapter 1.4, Problem 76E

a.

To determine

Verify the number of tax returns made through e-file.

Answer to Problem 76E

  $\boxed{6.5}$

Explanation of Solution

Given information:

E-Filing the table shows the numbers of tax returns (in millions) made through e-file from $2003$ through $2010$ . Let $f\left(t\right)$ represent the number of tax returns made through e-file in the year $t$ .

  

Find $\frac{f\left(2010\right)-f\left(2003\right)}{2010-2003}$ and interpret the result in the context of the problem.

Calculation:

  

  $\frac{f\left(2010\right)-f\left(2003\right)}{2010-2003}$

Consider the given table to interpret the result, from the table substitute $f\left(2010\right)=98.7$ and $f\left(2003\right)=52.9$ .

  $\begin{array}{l}\frac{f\left(2010\right)-f\left(2003\right)}{2010-2003}=\frac{98.7-52.9}{2010-2003}\\ =\frac{45.8}{7}\approx 6.5\end{array}$

Hence the rate of increase of number of tax returns made through e-file over the year $2003$ to $2010$ is $\boxed{6.5}$

b.

To determine

Verify the number of tax returns made through e-file.

Answer to Problem 76E

  

Explanation of Solution

Given information:

E-Filing The table shows the numbers of tax returns (in millions) made through e-file from $2003$ through $2010$ . Let $f\left(t\right)$ represent the number of tax returns made through e-file in the year $t$ .

  

Make a scatter plot of the data.

Calculation:

The scatter plot of the data is,

  

Hence the scatter plot of the data is plotted.

c.

To determine

Verify the number of tax returns made through e-file.

Answer to Problem 76E

  $\boxed{6.5}$

Explanation of Solution

Given information:

E-Filing the table shows the numbers of tax returns (in millions) made through e-file from $2003$ through $2010$ . Let $f\left(t\right)$ represent the number of tax returns made through e-file in the year $t$ .

  

Find a linear model for the data algebraically. Let $N$ represent the number of tax returns made through e-file and let $t=3$ correspond to $2003$ .

Calculation:

  

  $\frac{f\left(2010\right)-f\left(2003\right)}{2010-2003}$

Consider the given table to interpret the result, from the table substitute $f\left(2010\right)=98.7$ and $f\left(2003\right)=52.9$ .

  $\begin{array}{l}\frac{f\left(2010\right)-f\left(2003\right)}{2010-2003}=\frac{98.7-52.9}{2010-2003}\\ =\frac{45.8}{7}\approx 6.5\end{array}$

Hence the rate of increase of number of tax returns made through e-file over the year $2003$ to $2010$ is $\boxed{6.5}$

d.

To determine

Verify the number of tax returns made through e-file.

Answer to Problem 76E

  $\begin{array}{ccccccccc}t& 3& 4& 5& 6& 7& 8& 9& 10\\ N& 52.9& 59.4& 65.9& 72.4& 78.9& 85.4& 91.9& 98.4\end{array}$

Explanation of Solution

Given information:

E-Filing the table shows the numbers of tax returns (in millions) made through e-file from $2003$ through $2010$ . Let $f\left(t\right)$ represent the number of tax returns made through e-file in the year $t$ .

  

Use the model found in part $\left(c\right)$ to complete the table.

  

Calculation:

To find the $N$ substitute $t=3,4,5,6,7,8,9,10in$

  $N=6.5t+33.4$

  $\begin{array}{l}N=6.5\left(3\right)+33.4\\ =19.5+33.4\\ =52.9\end{array}$

  $\begin{array}{l}N=6.5\left(4\right)+33.4\\ =26+33.4\\ =59.4\end{array}$

  $\begin{array}{l}N=6.5\left(5\right)+33.4\\ =32.5+33.4\\ =65.9\end{array}$

  $\begin{array}{l}N=6.5\left(5\right)+33.4\\ =32.5+33.4\\ =65.9\end{array}$

  $\begin{array}{l}N=6.5\left(6\right)+33.4\\ =39+33.4\\ =72.4\end{array}$

  $\begin{array}{l}N=6.5\left(7\right)+33.4\\ =45.5+33.4\\ =78.9\end{array}$

  $\begin{array}{l}N=6.5\left(8\right)+33.4\\ =52+33.4\\ =85.4\end{array}$

  $\begin{array}{l}N=6.5\left(9\right)+33.4\\ =58.5+33.4\\ =91.9\end{array}$

  $\begin{array}{l}N=6.5\left(10\right)+33.4\\ =65+33.4\\ =98.4\end{array}$

Hence the table can be completed as,

  $\begin{array}{ccccccccc}t& 3& 4& 5& 6& 7& 8& 9& 10\\ N& 52.9& 59.4& 65.9& 72.4& 78.9& 85.4& 91.9& 98.4\end{array}$

e.

To determine

Verify the number of tax returns made through e-file.

Answer to Problem 76E

Results from part $\left(d\right)$ is less than results from actual data.

Explanation of Solution

Given information:

E-Filing the table shows the numbers of tax returns (in millions) made through e-file from $2003$ through $2010$ . Let $f\left(t\right)$ represent the number of tax returns made through e-file in the year $t$ .

  

Compare your results from part $\left(d\right)$ with the actual data..

Calculation:

By comparing the results from part $\left(d\right)$ and actual data from given table we conclude that the results from model is less than the actual data,

Hence, results from part $\left(d\right)$ is less than results from actual data.