Aria buys a new car for $34,104. She learns that every year her car will depreciate in value by 18%. Aria tracks how much her car is worth over time.

Question

Explain what the worth of Aria’s car will be compared to the previous year based on it’s depreciation rate. (1 pts)

Answer 1

Aria's car will depreciate in value by 18% each year. This means that at the end of each year, the car will be worth 82% (100% - 18%) of its value at the beginning of that year.

To find the worth of the car compared to the previous year, you can use the formula:

\[ \text{Value after year n} = \text{Initial value} \times (1 - \text{Depreciation rate})^n \]

For Aria's car, the value after one year would be:

\[ \text{Value after 1 year} = 34,104 \times 0.82 \]

Each subsequent year, you would multiply the value calculated from the previous year by 0.82 again. This results in the car's value decreasing consistently each year by 18% of its previous year's value. Thus, each year, the worth of Aria's car will be 82% of the previous year's worth, reflecting the effect of the annual depreciation rate.

Answer 2

Explain how the amount of money Aria’s car is worth over time is an example of exponential decay. (2 pts)

Answer 3

The amount of money Aria's car is worth over time is an example of exponential decay because its value decreases at a consistent percentage rate each year; in this case, 18%. This characteristic of consistently reducing by a fixed percentage results in a rapid decline initially, which slows over time but continues indefinitely.

Here's how it fits the concept of exponential decay:

1. Fixed Percentage Decrease: In exponential decay, the quantity decreases by a constant percentage rather than a fixed amount. For Aria's car, each year it retains only 82% of its value from the previous year, reflecting a consistent 18% depreciation. If the car were to depreciate by a fixed dollar amount each year, it would be linear decay, not exponential.

2. Mathematical Representation: The value of the car can be expressed using an exponential function. The formula that represents the car's value over time is:

   \[ V(t) = V_0 \times (1 - r)^t \] where \(V(t)\) is the value after \(t\) years, \(V_0\) is the initial value ($34,104), \(r\) is the depreciation rate (0.18), and \(t\) is the time in years. This formula indicates that the remaining value decreases exponentially as time increases.

3. Graphical Representation: If you were to graph the value of the car over time, you'd see a curve that starts relatively high and decreases sharply at first, then levels off as the years go by. This downward curve demonstrates the nature of exponential decay, showcasing that while the value continues to decline, the rate of decrease slows down as the value approaches zero.

In summary, Aria's car's depreciation exemplifies exponential decay because it decreases by a consistent percentage each year, as modeled by an exponential function, leading to a gradual, sustained decline in value over time.