Chapter 3.3, Problem 19PPS

a.

To determine

The linear function to model the price of the car with respect to age.

Answer to Problem 19PPS

  $y=-1811.67x+19548.3$

Explanation of Solution

Given:

The table:

  

Concept Used:

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

Calculation:

From the given table is clear that the value of the car with age 2 years is 15924.96 dollars and the value of the car with age 3 years is 14113.29 dollars.

The rate of change of a line between two points is given by the slope of the line.

So, the slope of the linear function to model the price of the car with respect to age is

  $\begin{array}{c}\text{slope, }m=\frac{14113.29-15924.96}{3-2}\\ =\frac{-1811.67}{1}\\ =-1811.67\end{array}$

So, the equation of the line passing through $\left(2,15924.96\right)$ and having slope $-1811.67$ is

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

  $\begin{array}{c}y-15924.96=-1811.67\left(x-2\right)\end{array}$

  $\begin{array}{c}y-15924.96=-1811.67x+3623.34\end{array}$

        $\begin{array}{c}y=-1811.67x+3623.34+15924.96\end{array}$

        $\begin{array}{c}y=-1811.67x+19548.3\end{array}$

Thus, the linear function that models the price of the car with respect to age is $y=-1811.67x+19548.3$ .

b.

To determine

To interpret the meaning of the slope of the line.

Answer to Problem 19PPS

The value of the car is decreasing at a rate of $1811.67$ dollars per year.

Explanation of Solution

Given:

The table:

  

Concept Used:

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

Calculation:

The slope of the line is $-1811.67$ , it implies that the value of the car is decreasing at a rate of $1811.67$ dollars per year.

c.

To determine

The average retail price for a 7-year old car.

Answer to Problem 19PPS

  $6866.61$ dollars

Explanation of Solution

Given:

The table:

  

Concept Used:

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

Calculation:

In order to find the average retail price for a 7-year old car, substitute $x=7$ in the linear function $y=-1811.67x+19548.3$ and simplify further as shown below,

  $\begin{array}{c}y=-1811.67x+19548.3\end{array}$

  $\begin{array}{c}y=-1811.67\left(7\right)+19548.3\end{array}$

  $\begin{array}{c}y=-12681.69+19548.3\end{array}$

  $\begin{array}{c}y=6866.61\end{array}$

Thus, the average retail price for a 7-year old car is $6866.61$ dollars.