The average cost of a new car has been changing approximately linearly with time. In 1990, the average cost was $16,950, and the average cost was $27,957 in2008

Question

a.) Let x = the number of years after
1990 and let y = average cost of
a new car. What ordered pairs
can be written to express this
information?

b) Write the equation of the line
through these two points in
point-slope form.

 

c) Convert the equation of the line
to slope-intercept form and
explain in your own words what
the slope means in the context
of this problem.

 

d) Based on this linear model, what
will be the average cost of a new
car in 2018?

Answer 1

a)

 

Time is always in order from past to present.  So we must list the order pairs in that order.  Since x is the number of years after 1990, then x=0 represent 1990, x=1 represents 1991, x=2 represents 1992, etc.

 

Order pair one = (0, 16950)

Order pair two = (18, 27957)

 

b)

 

We can use these ordered pairs to write the equation.  Since this is linear, we can find the slope using the order pairs.

 

slope = (y₂ - y₁) / (x₂ - x₁)

slope = (27957 - 16950) / (18 - 0)

slope = 611.50

 

The slope represent the average cost of a new car per year.

 

 

So far, the equation looks like

 

y = 611.50x + b

 

 

Next, we will plug in ordered pair two into the equation to find b.

 

27957 = 611.50(18) + b

27957 = 11007 + b

16950 = b

 

The equation of the line is

 

y = 611.50x + 16950

 

 

c)

 

Use part b to find the cost in 2018.

 

x = 28

 

y = 611.50(28) + 16950

y = 34072

 

 

The average cost of a new car in 2018 is $34,072.