Aria buys a new car for $__________. She learns that every year her car will

Question

depreciate in value by 15%. This means that each year her car will be worth 85% of
what it was the previous year. Aria tracks how much her car is worth over time.

A. Choose a dollar amount that you think Aria might have purchased her car for.

B. Explain how the amount of money Aria’s car is worth over time is an example
of exponential decay.

C. Create a sequence that describes the worth of Aria’s car each year for 10
years.

D. Create a formula for the worth of Aria’s car over time.

E. On a sheet of graph paper, create a graph for the worth of Aria’s car over time.
Describe at least one key feature on your graph that signals it depicts
exponential decay.

Simplify your answers, shorten them as much as possible.
Answer each question seperately.

Answer 1

A.

Aria buys a new car for $30,000.

B.

Aria's car value decreasing by 15% each year illustrates exponential decay because the value reduces by a consistent percentage rather than a fixed amount. Consequently, the car's worth diminishes over time at a rate proportional to its current value, leading to a declining curve where the quantity decreases quickly at first and then slows down as it approaches zero.

C.

Sequence for Aria’s car worth over 10 years:

1. Year 0: $30,000
2. Year 1: $30,000 * 0.85 = $25,500
3. Year 2: $25,500 * 0.85 = $21,675
4. Year 3: $21,675 * 0.85 = $18,423.75
5. Year 4: $18,423.75 * 0.85 = $15,661.19
6. Year 5: $15,661.19 * 0.85 = $13,319.01
7. Year 6: $13,319.01 * 0.85 = $11,317.16
8. Year 7: $11,317.16 * 0.85 = $9,621.58
9. Year 8: $9,621.58 * 0.85 = $8,179.34
10. Year 9: $8,179.34 * 0.85 = $6,952.43
11. Year 10: $6,952.43 * 0.85 = $5,908.56

D.

Formula for the worth of Aria’s car over time:
\[ V(t) = 30000 \times (0.85)^t \]
Where \( V(t) \) is the value of the car after \( t \) years.

E.

On a sheet of graph paper, plot the sequence values for each year from 0 to 10.

1. Key feature: The graph should show a downward curve that flattens as time increases, indicating that the car’s value is decreasing at a decreasing rate, characteristic of exponential decay. The slope of the curve is steeper initially and becomes less steep over time, demonstrating that the depreciation is slowing as the car’s worth approaches zero.