Chapter 9.7, Problem 3P

To determine

Find the type of function that best models the data.Write an equation to model the data.

Answer to Problem 3P

Linear model ; $y=480x+18254$

Explanation of Solution

Given :

  

The table shows the annual income of a small theater company.

Calculation:

Let x represent the year.

y represent the income.

First we plot the points on the graph.

  

So, the points are in a line.

We check the difference in points of x and in points of y :

Difference in x -coordinates	Difference in y -coordinates
1-0=1	18730-18254=476
2-1=1	19215-18730=485
3-2=1	19695-19215=480
4-3=1	20175-19695=480

The y-values have common difference of about 480.

So, the model can be represented by the linear equation of the form $y=mx+b$

The slope is $m=\frac{480}{1}=480$ .

Consider the point (0,18254) and substitute in the equation to find b :

  $\begin{array}{l}y=mx+b\\ \Rightarrow 18254=480(0)+b\\ \Rightarrow b=18254\end{array}$

So, the equation that models the given data is :

  $y=480x+18254$

Check the equation by substituting any point in it.

  $\begin{array}{l}y=480x+18254\\ \Rightarrow 480(1)+18254=18734\approx 18730=y\mathrm{.....}[(x,y)=(1,18730)]\end{array}$

So, the equation is correct.