To determine the value of a car after 6 years, given an initial purchase price and a constant annual depreciation rate, we'll follow these steps:
1. Identify the key parameters:
- Initial value of the car: \(\( \$15,500 \)\)
- Annual depreciation rate: \(\( 7.5\% \)\) (or 0.075 in decimal form)
- Number of years: \(\( 6 \)\)
2. Understanding Depreciation:
Depreciation means that the car's value decreases by a certain percentage each year. For this problem, it decreases by \(\( 7.5\% \)\) every year. The depreciation process can be represented by multiplying the car's value by \(\( (1 - \text{depreciation rate}) \)\).
Therefore, each year, the car's value becomes \(\( \text{value} \times (1 - 0.075) \)\).
3. Formula for Depreciation:
For constant percentage depreciation, the formula to find the car's value after \(\( n \)\) years is:
\(\[
\text{Final Value} = \text{Initial Value} \times (1 - \text{Depreciation Rate})^n
\]\)
Here, \(\( n \)\) is the number of years.
4. Substitute the Known Values:
\(\[
\text{Final Value} = 15500 \times (1 - 0.075)^6
\]\)
5. Calculate Step by Step:
- First, compute \(\( 1 - \text{Depreciation Rate} \)\):
\(\[
1 - 0.075 = 0.925
\]\)
- Then, raise this result to the power of 6 (number of years):
\(\[
0.925^6
\]\)
- Finally, multiply by the initial value:
\(\[
\text{Final Value} = 15500 \times 0.925^6
\]\)
6. Perform the Calculation:
- Compute \(\( 0.925^6 \)\):
\(\[
0.925^6 \approx 0.6353
\]\)
- Multiply by the initial value:
\(\[
15500 \times 0.6353 \approx 9857.15
\]\)
7. Rounding to the Nearest Cent:
The exact value after multiplication is approximately \(\( \$9857.145 \)\). When rounded to the nearest cent, this becomes \(\( \$9857.15 \)\).
So, after 6 years, the value of the car will be approximately \$9857.15.
