Question

1. tristan is looking for a new car. he has particular interest in an expensive sports car with a list price of $32,500. tristan knows that the minute he drives the car off the lot, it will start to lose value, or depreciate. he finds out that the car will depreciate to a scrap value of $1000 in 15 years.

a. what is the total change in value of the car in 15 years?
b. what is the average amount of depreciation per year?
c. when the car is exactly 7.5 years old, tristan decides to sell it to his friend jovann. what is the value of the car when jovann buys it?
d. what are the two quantities that are changing in part (c)? define and identify the independent and dependent variables for the quantities you defined with their units of measure.
e. write an equation to calculate the value of the car given the number of years jovann has owned the car.
f. create a table of values that includes when jovann has owned the car 0 years, 6 months, and two and half years. also, include when the value of the car was $1000, $22,000, $25,150, $30,400, and $32,500.
g. create a graph of the data from the table. carefully select the lower bound, upper bound, and intervals. remember to label the axes and the intervals.
h. locate the point where the value of the independent quantity is -5. what is the value of the dependent quantity at this point? write the point as an ordered pair. what does the ordered pair mean in the context of the problem?

Explanation:

Step1: Calculate total change in value

The initial value is $32500$ and the final value is $1000$. The total change in value is $32500 - 1000$.
$$32500-1000 = 31500$$

Step2: Calculate average depreciation per year

Total change is $31500$ over $15$ years. The average depreciation per year is $\frac{31500}{15}$.
$$\frac{31500}{15}=2100$$

Step3: Calculate value at 7.5 years

The car depreciates at $2100$ per year. After $7.5$ years, the depreciation is $2100\times7.5$. The value is $32500-2100\times7.5$.
$$2100\times7.5 = 15750$$
$$32500 - 15750=16750$$

Step4: Identify variables

The two changing quantities are time (in years) and value (in dollars). The independent variable is time (years) and the dependent variable is value (dollars) as the value depends on the time.

Step5: Write the equation

Let $y$ be the value of the car and $x$ be the number of years since Tristan bought the car. The equation is $y=32500 - 2100x$

Step6: Create table (partial - for 0 years, 0.5 years, 2.5 years)

Years ($x$)	Value ($y$)
$0.5$	$32500-2100\times0.5=32500 - 1050=31450$
$2.5$	$32500-2100\times2.5=32500-5250 = 27250$

Step7: For $x=-5$

Substitute $x =- 5$ into $y=32500-2100x$
$$y=32500-2100\times(-5)=32500 + 10500=43000$$
The ordered pair is $(-5,43000)$. In the context of the problem, it means 5 years before Tristan bought the car (a hypothetical situation), the car would have a value of $43000$ (assuming the same linear depreciation model holds retroactively)

Answer:

a. The total change in value is $\$31500$.
b. The average amount of depreciation per year is $\$2100$.
c. The value of the car when Jovann buys it is $\$16750$.
d. The two changing quantities are time (independent, years) and value (dependent, dollars).
e. The equation is $y = 32500-2100x$.
f. (Table as above - can be expanded as per requirements).
g. (Graph - x - axis: years, y - axis: value. Lower bound for x: 0, upper bound: 15 (or as per data in table). Lower bound for y: 1000, upper bound: 32500. Intervals for x: 1 year, for y: 1000 or 2000).
h. The ordered pair is $(-5,43000)$. It means 5 years before Tristan bought the car (hypothetically), the car's value is $\$43000$.