Aria's car is an example of exponential decay because Aria's car is losing value each year by a factor of 85% (or a decay factor of 0.85).
To create a sequence that describes the worth of Aria's car each year for 8 years, we start with the initial value of $36,000 and multiply by 0.85 (the remaining value after depreciation) for each subsequent year.
- Year 0: \( 36,000 \)
- Year 1: \( 36,000 \times 0.85 = 30,600.00 \)
- Year 2: \( 30,600 \times 0.85 = 26,010.00 \)
- Year 3: \( 26,010 \times 0.85 = 22,108.50 \)
- Year 4: \( 22,108.50 \times 0.85 = 18,791.23 \)
- Year 5: \( 18,791.23 \times 0.85 = 15,973.54 \)
- Year 6: \( 15,973.54 \times 0.85 = 13,579.51 \)
- Year 7: \( 13,579.51 \times 0.85 = 11,528.58 \)
- Year 8: \( 11,528.58 \times 0.85 = 9,791.29 \)
Now we can write the sequence of values for each year:
- Year 0: $36,000.00
- Year 1: $30,600.00
- Year 2: $26,010.00
- Year 3: $22,108.50
- Year 4: $18,791.23
- Year 5: $15,973.54
- Year 6: $13,579.51
- Year 7: $11,528.58
- Year 8: $9,791.29
Next, we can create a formula for the worth of Aria's car over time using the exponential decay formula \(y = a \cdot b^x\).
Where:
- \(a = 36,000\) (initial amount)
- \(b = 0.85\) (decay factor)
- \(x\) is the number of years
Thus, the formula becomes:
\[ y = 36,000 \cdot (0.85)^x \]
Now, let’s use this formula to find out how much Aria's car is worth in 20 years:
\[ y = 36,000 \cdot (0.85)^{20} \]
First, calculate \( (0.85)^{20} \):
\[ (0.85)^{20} \approx 0.038759 \]
Now, substitute this value back into the formula:
\[ y \approx 36,000 \cdot 0.038759 \approx 1395.24 \]
So, after 20 years, Aria's car will be worth approximately $1,395.24.