Question

(10 points), Joe bought a new car in 2011 for $45,000. In 2015, Joe was offered a fair price of$37,000 for his car, but he turned it downa) Build a linear algebraio model, (ie, a function), that helps Joe find the car's value when itis 7 years oldb) Use your model to give an estimate to the current value of Joe's car?c) Use your model to give an estimate to the value of Joe's car in 2012

Answer 1

Line equation

STEP 1: graph of the given points

We have two given values:

In 2011 the car costs $45,000

In 2015, the car costs $37,000.

Let's graph this function, the x-axis is going to be the years and the y- axis the cost.

Then, we are going to locate y=45000 at x=2011:

(2011, 45000)

and y=37000 at x=2015:

(2015, 37000).

STEP 2: line graph

We want to find a function for the straight line that connects the points:

STEP 3: line equation

We know that the equation of a line is given by:

y =mx + b

where m and b are numbers: m is the slope (it shows the inclination of the line) and b is the y-intercept.

We want to find m and b to obtain the equation.

m: slope

We have two points (first step):

(x₁, y₁) = (2011, 45000)

(x₂, y₂) = (2015, 37000)

We find the change of each variable from point 1 to point 2:

Δx = x₂ - x₁ = 2015 - 2011

= 4

Δy = y₂ - y₁ = 37000 - 45000

= -8000

We have that the slope is given by the rate between Δy and Δx:

`\begin{gathered} m=(\Delta y)/(\Delta x)=(y_(2)-y_(1))/(x_(2)-x_(1)) \\ m=-(8000)/(4)=-2000 \end{gathered}`

Then m = -2000, so this line equation should look like:

y =mx + b

↓

y = -2000x + b

b: y-intercept

In order to find b we replace x and y in the equation by a value we know they should have.

We have that when it is x=2011, then y=45000. Let's replace:

y = -2000x + b

↓

45000= -2000 · 2011 + b

Let's solve the equation for b:

45000= -2000 · 2011 + b

↓since -2000 · 2011 = -4,022,000

45,000= -4,022,000 + b

↓ taking -4,022,000 to the left

45,000 +4,022,000 =b

4,067,000=b

Then b= 4,067,000, so this line equation should look like:

y = -2000x + b

↓

y = -2000x + 4,067,000

Answer A: a linear model is y = -2000x + 4,067,000

STEP 4: estimate to the current value of Joe's car

This is the year 2021. This means that x=2021. Let's replace in the equation and see what happens with y (the price of the car) this year:

y = -2,000x +4,067,000

↓

y = -2,000 ·2021 +4,067,000

Since -2,000 ·2021 = -4,042,000

y = -4,042,000 +4,067,000

y = 25,000

This means that the current value of the car is $25,000

Answer B: the current value of Joe's car is $25,000

STEP 5: estimate to the value of Joe's car in 2012

In 2012, x=2012. Replacing in the equation:

y = -2,000x +4,067,000

↓

y = -2,000 ·2012 +4,067,000

Since -2,000 ·2012 = -4,024,000

y = -4,024,000 +4,067,000

y = 43,000

Answer C: in 2012 the value of Joe's car was $43,000