Chapter 2.7, Problem 79E

a.

To determine

To create a scatter plot of the data by using a graphing utility.

Answer to Problem 79E

  

Explanation of Solution

Given information:

The number $N$ (in millions) of students enrolled in schools in the United States from $2000$ through $2009$ are shown in the table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

Use a graphing data utility to create a scatter plot of the data. Let $t$ represent the year, with $t=0$ corresponding to $2000$ .

Calculation:

Here, we will consider the following table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

  

Hence, the result is shown in the graph.

b.

To determine

To find a quartic model for the data by using regression feature of the graphing utility.

Answer to Problem 79E

  $\boxed{N=0.00406t^4-0.0564t^3+0.147t^2+0.86t+72.2}$

Explanation of Solution

Given information:

The number $N$ (in millions) of students enrolled in schools in the United States from $2000$ through $2009$ are shown in the table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

Use the regression feature of the graphing utility to find a quartic model for the data. (A quartic model has the form $a{t}^4+b{t}^3+c{t}^2+dt+e$ , where $a,b,c,d$ and $e$ are constant and $t$ is variable.)

Calculation:

By using the regression feature of the graphing utility a quartic model for the data is given by:

  $\boxed{N=0.00406t^4-0.0564t^3+0.147t^2+0.86t+72.2}$ .

Hence, the result is $\boxed{N=0.00406t^4-0.0564t^3+0.147t^2+0.86t+72.2}$ .

c.

To determine

How well does the model fit the data?

Answer to Problem 79E

  

Explanation of Solution

Given information:

The number $N$ (in millions) of students enrolled in schools in the United States from $2000$ through $2009$ are shown in the table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

Graph the model and the scatter plot in the same viewing window. How well does the model fit the data?

Calculation:

The graph of the model on the same scatter plot is shown below:

  

Hence, from the above graph we can observe that the model fits the data very closely.

d.

To determine

To find when did the number of students enrolled in the schools exceed $74$ million?

Answer to Problem 79E

  $\boxed{Year2001}$

Explanation of Solution

Given information:

The number $N$ (in millions) of students enrolled in schools in the United States from $2000$ through $2009$ are shown in the table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

According to the model, when did the number of students enrolled in the schools exceed $74$ million?

Calculation:

Hence, by using the zoom and trace features, the number of students enrolled in schools exceeded $74$ million in the year $\boxed{2001}$ .

e.

To determine

To find if the model is valid for long-term predictions of student enrollment?

Answer to Problem 79E

No, the model is not valid.

Explanation of Solution

Given information:

The number $N$ (in millions) of students enrolled in schools in the United States from $2000$ through $2009$ are shown in the table:

  $\begin{array}{ll}Year\hfill & Number,N\hfill \\ 2000\hfill & 72.2\hfill \\ 2001\hfill & 73.1\hfill \\ 2002\hfill & 74.0\hfill \\ 2003\hfill & 74.9\hfill \\ 2004\hfill & 75.5\hfill \\ 2005\hfill & 75.8\hfill \\ 2006\hfill & 75.2\hfill \\ 2007\hfill & 76.0\hfill \\ 2008\hfill & 76.3\hfill \\ 2009\hfill & 77.3\hfill \end{array}$

Is the model valid for long-term predictions of student enrollment? Explain.

Calculation:

No, the model is not valid for long-term predictions of student enrollment. As, the numbers predicted by the model increase very fast while estimated far into the future which may not be correct.

Hence, the model is not valid for long-term predictions of student enrollment.