Q 77. - Precalculus Enhanced with Graphing Utilities

Question

National Debt The size of the total debt owed by the United States federal government has been growing over the past few years. In fact, according to the Department of the Treasury, the debt per person living in the United States is approximately $\backslash (45,000$ (or over $\backslash )300,000$ per U.S. household).

The data below represent the U.S. debt for the years $2000–2010$. Since the debt D depends on the year $y$ and each input corresponds to exactly one output, the debt is a function of

the year; so $D\left(y\right)$ represents the debt for each year $y$.

Year	Debt(Billion of Dollars)
$2000$	$5674$
$2001$	$5807$
$2002$	$6228$
$2003$	$6783$
$2004$	$7379$
$2005$	$7933$
$2006$	$8507$
$2007$	$9008$
$2008$	$10,025$
$2009$	$11,910$
$2010$	$13,562$

(a) Plot the points $(2000, 5.7), (2001, 5.8),$ and so on in a Cartesian plane.

(b) Draw a line segment from the point $(2000, 5.7)$ to $(2002, 6.2)$. What does the slope of this line segment represent?

(c) Find the average rate of change of the debt from $2000$ to$2002$.

(d) Find the average rate of change of the debt from $2004$ to $2006$.

Step-by-Step Solution

Answer

Part (a). The required graph is shown below:

Part (b). The graph with the line segment is shown below:

The slope of the line segment represents the average rate of change of the debt from the year $2000$ to $2002$.

Part (c). The average rate of change from $2000$ to $2002$ is $\$227$ billion.

Part (d). The average rate of change from $2004$ to $2006$ is $\$564$ billion.

Part (e). The average rate of change from $2008$ to $2010$ is $\$1768.5$ billion.

Part (f). The average rate of change of the debt is increasing as time passes. 

Part (a). Step 1. Given information

The data that represents the U.S. debt for the years $2000–2010$ is shown in the table below:

Year	Debt(Billion of Dollars)
$2000$	$5674$
$2001$	$5807$
$2002$	$6228$
$2003$	$6783$
$2004$	$7379$
$2005$	$7933$
$2006$	$8507$
$2007$	$9008$
$2008$	$10,025$
$2009$	$11,910$
$2010$	$13,562$

Part (a). Step 2. Plot the points ( 2000 ,   5 . 7 ) , ( 2001 ,   5 . 8 ) , and so on in a Cartesian plane.

The required graph is shown below:

Part (b) Step 1. Draw a line segment from the point ( 2000 ,   5 . 7 ) to ( 2002 ,   6 . 2 ) to determine the representation of the slope of the line segment.

The graph with the line segment from the point $(2000, 5.7)$ to $(2002, 6.2)$ is shown below:

From the graph, observe that the slope of the line segment represents the average rate of change of the debt from the year $2000$ to $2002$.

Part (c). Step 1. Determine the average rate of change of the debt from the year 2000 to 2002 .

The average rate of change from $a$ to $b$ is defined as folllows:

$\frac{∆y}{∆x}=\frac{f\left(b\right)-f\left(a\right)}{b-a}\dots \left(1\right)$, where $a\ne b$.

Substitute $a=2000$ and $b=2002$ into $\left(1\right)$.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{f\left(2002\right)-f\left(2000\right)}{2002-2000}\end{array}$

Substitute $f\left(2000\right)=5674$ and $f\left(2002\right)=6228$ from the given table.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{6228-5674}{2}\\ & =& \frac{554}{2}\\ & =& 227\end{array}$

Thus, the average rate of change of the debt from the year $2000$ to $2002$ is $\$227$ billion.

Part (d). Step 1. Determine the average rate of change of the debt from the year 2004 to 2006 .

Substitute $a=2004$ and $b=2006$ into $\left(1\right)$.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{f\left(2006\right)-f\left(2004\right)}{2004-2002}\end{array}$

Substitute $f\left(2004\right)=7379$ and $f\left(2006\right)=8507$ from the given table.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{8507-7379}{2}\\ & =& \frac{1128}{2}\\ & =& 564\end{array}$

Thus, the average rate of change of the debt from the year$2004$ to $2006$ is $\$564$ billion.

Part (e). Step 1. Determine the average rate of change of the debt from the year 2008 to 2010 .

Substitute $a=2008$ and $b=2010$ into .

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{f\left(2010\right)-f\left(2008\right)}{2010-2008}\end{array}$

Substitute $f\left(2008\right)=5674$ and $f\left(2010\right)=6228$ from the given table.

$\begin{array}{rcl}\frac{∆y}{∆x}& =& \frac{13,562-10,025}{2}\\ & =& \frac{3537}{2}\\ & =& 1768.5\end{array}$

Thus, the average rate of change of the debt from the year $2008$ to $2010$ is $\$1768.5$ billion.

Part (f) Step 1. Determine the state to the average rate of change as time passes.

The average rate of change of the debt is increasing as time passes.