Q32. - Algebra 2

Question

For exercises 30-32, use the table that shows the amount Americans spent on recreation in recent years.

32.    Predict the amount that will be spent on recreation in 2010.

Step-by-Step Solution

Answer

The predicted amount that will be spent on recreation in 2010 is $821.6.

Step-1 – Apply the concept of slope-intercept form of a linear equation, point slope form, prediction equation

The slope of a line is the ratio of the change in the y-coordinates to the change in the x- coordinates.

Symbolically, the slope m of the line passing through the points $\left(x_1,y_1\right)\text{ and }\left(x_2,y_2\right)$ is given by $m=\frac{y_2-y_1}{x_2-x_1}$ where $x_1\ne x_2$.

 

The slope-intercept form of the equation of a line is given by $y=mx+b$ where m is the slope and b is the y-intercept.

 

The point slope form of a equation of a line is given by $y-y_1=m(x-x_1)$ where $\left(x_1,y_1\right)$ are the coordinates of a point on the line and m is the slope of the line.

 

A prediction equation is an equation which is used to predict one of the variables given the other variable.

In order to find a prediction equation for a set of data, select two points that appear to represent the data well.

Step-2 – Find the ordered pairs

Let x  represents the number of years since 1995. Therefore take the number of years since 1995 on the horizontal axis and amount on the vertical axis.

From the table find the ordered pairs which are as follows:

$\left(1995,401.6\right),\left(1996,429.6\right),\left(1997,457.8\right),\left(1998,494.7\right)$

Step-3 – Draw the scatter plot

The scatter plot is as shown below:

Step-4 – Find the slope

A scatter plot suggests that any two points can be used to find a prediction equation. Hence use $\left(1995,401.6\right)\text{ and }\left(1996,429.6\right)$.

Therefore, the slope is obtained as follows:

$\begin{array}{c}m=\frac{y_2-y_1}{x_2-x_1}\\ =\frac{429.6-401.6}{1996-1995}\\ =\frac{28}{1}\\ =28\end{array}$

Step-5 – Find the equation in slope intercept form

By using the value of slope in point slope form the equation is obtained in slope intercept form as follows:

$\begin{array}{c}y-y_1=m\left(x-x_1\right)\\ y-401.6=28\left(x-1995\right)\\ y-401.6=28x-\left(28\right)1995\\ y-401.6=28x-\left(28\right)1995\\ y-401.6=28x-55860\end{array}$

Keep the variable y on one side and the rest on other side. Hence,

$\begin{array}{c}y-401.6+401.6=28x-55860+401.6\text{ (adding 401.6 on both sides)}\\ y=28x-55458.4\end{array}$

Hence the equation is obtained in slope intercept form and this is one prediction equation.

Step-6 – Predict the amount

In order to predict the amount that will be spent on recreation in 2010, take the value of x in the prediction equation as 2010. Therefore,

$\begin{array}{c}y=28x-55458.4\\ \Rightarrow y=28\left(2010\right)-55458.4\\ \Rightarrow y=56280-55458.4\\ \Rightarrow y=821.6\end{array}$

Hence the predicted the amount that will be spent on recreation in 2010 is $821.6.