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