Question

dylan wants to compare the resale value over time for the toyota 4 - runner and the jeep cherokee.
toyota 4 - runner

age	resale value
2	34,047
3	33,075
4	30,062
5	29,266
6	26,781
7	24,526
8	21,404
9	19,393
10	18,783

jeep cherokee
(there is a graph with points (1,24630) and (4,19491) and a line representing resale value over age)
a. which vehicle had a higher sales price when brand new (age = 0)? justify your answer using mathematics.
b. determine the average rate of change in resale value from 1 to 4 years for each car. which vehicle is losing its resale value faster?
c. which vehicle will reach a resale value of $5000 first? justify your answer using mathematics.

Explanation:

Step1: Find initial - value for Toyota 4 - Runner

Assume the resale value follows a linear - like depreciation. For the Toyota 4 - Runner, we can use the two - point form of a line $y - y_1=\frac{y_2 - y_1}{x_2 - x_1}(x - x_1)$. Let $(x_1,y_1)=(1,35866)$ and $(x_2,y_2)=(2,34047)$. The slope $m=\frac{34047 - 35866}{2 - 1}=-1819$. Then the equation of the line is $y-35866=-1819(x - 1)$. When $x = 0$, $y=35866+1819 = 37685$.
For the Jeep Cherokee, using the two points $(1,24630)$ and $(4,19491)$ on the graph. The slope $m=\frac{19491 - 24630}{4 - 1}=\frac{-5139}{3}=-1713$. The equation of the line is $y - 24630=-1713(x - 1)$. When $x = 0$, $y=24630 + 1713=26343$. Since $37685>26343$, the Toyota 4 - Runner had a higher sales price when brand new.

Step2: Calculate average rate of change for Toyota 4 - Runner

The average rate of change formula is $\frac{\Delta y}{\Delta x}=\frac{y_2 - y_1}{x_2 - x_1}$. For the Toyota 4 - Runner, $x_1 = 1,y_1 = 35866,x_2 = 4,y_2 = 30062$. The average rate of change is $\frac{30062-35866}{4 - 1}=\frac{-5804}{3}\approx - 1934.67$.

Step3: Calculate average rate of change for Jeep Cherokee

For the Jeep Cherokee, with $x_1 = 1,y_1 = 24630,x_2 = 4,y_2 = 19491$. The average rate of change is $\frac{19491 - 24630}{4 - 1}=\frac{-5139}{3}=-1713$. Since $| - 1934.67|>| - 1713|$, the Toyota 4 - Runner is losing its resale value faster from 1 to 4 years.

Step4: Find time for Toyota 4 - Runner to reach $5000

Using the equation $y=37685-1819x$ (from Step 1), set $y = 5000$. Then $5000=37685-1819x$. Solving for $x$ gives $1819x=37685 - 5000=32685$, so $x=\frac{32685}{1819}\approx17.97$.

Step5: Find time for Jeep Cherokee to reach $5000

Using the equation $y=26343-1713x$ (from Step 1), set $y = 5000$. Then $5000=26343-1713x$. Solving for $x$ gives $1713x=26343 - 5000=21343$, so $x=\frac{21343}{1713}\approx12.46$. Since $12.46<17.97$, the Jeep Cherokee will reach a resale value of $5000 first$.

Answer:

a. The Toyota 4 - Runner had a higher sales price when brand new.
b. The average rate of change for the Toyota 4 - Runner from 1 to 4 years is approximately - 1934.67, and for the Jeep Cherokee is - 1713. The Toyota 4 - Runner is losing its resale value faster.
c. The Jeep Cherokee will reach a resale value of $5000 first$.