Aria buys a new car for $36,000. She learns that every year her car will 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.
Explain how the amount of money Aria’s car is worth over time is an example of exponential decay.
Create a sequence that describes the worth of Aria’s car each year for 8 years.
Create a formula for the worth of Aria’s car over time
y = a⋅b^x
a= initial amount
b= growth factor
Use your formula in C and find how much Aria’s car is worth in 20 years
The amount of money Aria's car is worth over time is an example of exponential decay because the value of her car decreases by a constant rate of 15% each year. This means that the rate of depreciation remains the same over time, resulting in a graph that is curved and gets steeper as time goes on.
To create a sequence that describes the worth of Aria's car each year for 8 years, we start with the initial amount of $36,000 and multiply it by 0.85 (the factor of 85%) each year.
Year 1: $36,000 * 0.85 = $30,600
Year 2: $30,600 * 0.85 = $26,010
Year 3: $26,010 * 0.85 = $22,108.50
Year 4: $22,108.50 * 0.85 = $18,792.23
Year 5: $18,792.23 * 0.85 = $15,973.40
Year 6: $15,973.40 * 0.85 = $13,577.39
Year 7: $13,577.39 * 0.85 = $11,541.78
Year 8: $11,541.78 * 0.85 = $9,810.51
The formula for the worth of Aria's car over time can be expressed as:
y = a * b^x
Where:
y = worth of Aria's car at a given year x
a = initial amount (in this case $36,000)
b = growth factor (in this case 0.85, representing the 15% depreciation rate each year)
x = year
To find how much Aria's car is worth in 20 years, we substitute the values into the formula:
y = 36000 * (0.85)^20
Using a calculator, we find that Aria's car will be worth approximately $4,124.38 in 20 years.