Beginning Algebra With Applications, Chapter 5, 5.4, Section 5.4, Problem 72

The table below shows the median weekly salary of a union member for various years.

$\begin{array}{|c|c|} \hline\\ \text{Year} & \text{Weekly Salary} \\ & \text{(in dollars)} \\ \hline\\ 2000 & 696 \\ \hline\\ 2001 & 718 \\ \hline\\ 2002 & 740 \\ \hline\\ 2003 & 760 \\ \hline\\ 2004 & 781 \\ \hline \end{array}$

a. Determine the equation of the line between (2000, 696) and (2004, 781).

We let $(x_1, y_1) = (2000,696)$ and $(x_2, y_2) = (2004, 781)$

Using the slope of the line,


$\begin{equation} \begin{aligned} m =& \frac{y_2 - y_1}{x_2 - x_1} \\ \\ m =& \frac{781-696}{2004-2000} \\ \\ m =& \frac{85}{4} \end{aligned} \end{equation}$


Using the Point Slope Formula, where $\displaystyle m = \frac{85}{4}$ and $(x_1, y_1) = (2000,696)$


$\begin{equation} \begin{aligned} y - y_1 =& m(x - x_1) && \\ \\ y-696 =& \frac{85}{4} (x-2000) && \text{Substitute } m = \frac{85}{4} \text{ and } (x_1, y_1) = (2000,696) \\ \\ y - 696 =& \frac{85}{4}x - 500 && \text{Apply Distributive Property} \\ \\ y =& \frac{85}{4}x - 500 + 696 && \text{Simplify} \\ \\ y =& \frac{85}{4}x + 196 && \end{aligned} \end{equation}$


b. What was the average annual rate of change in the median weekly salary for a union employee between 2000 and 2004?


$\begin{equation} \begin{aligned} \text{average rate of change} =& \frac{\text{weekly salary from 2004 - weekly salary from 2000}}{2004-2000} \\ \\ =& \frac{781-696}{2004-2000} \\ \\ =& \frac{85}{4} \\ \\ =& 21.25 \end{aligned} \end{equation}$


The average annual rate of change in the median weekly salary for a union employee between 2000 and 2004 is $\$ 21.25$.

c. Suppose the trend shown by the equation in part a were to continue, what would be the median weekly salary of a union worker in 2012?

Since the rate of change is linear, the slope is the same for any points. So we let $(x_1, y_1) = (2000, 696)$ and $(x_2, y_2) = (2012,n)$, $\displaystyle m = \frac{85}{4}$, then


$\begin{equation} \begin{aligned} \frac{85}{4} =& \frac{n-696}{2012-2000} && \\ \\ \frac{85}{4} =& \frac{n-696}{12} && \text{Multiply both sides by } 12 \\ \\ 255 =& n-696 && \text{Add } 696 \\ \\ n =& 951 && \end{aligned} \end{equation}$


The median weekly salary of a union worker in 2012 is $\$ 951$.