To find the approximate value of Doug's car after [tex]\( 4 \frac{1}{2} \)[/tex] years, we will use the given formula:
[tex]\[ V = P(1 - r)^t \][/tex]
### Step-by-Step Solution:
1. Identify the given variables:
- [tex]\( P = 25000 \)[/tex] (initial price of the car)
- [tex]\( r = 0.20 \)[/tex] (annual rate of depreciation)
- [tex]\( t = 4.5 \)[/tex] (number of years)
2. Substitute the variables into the formula:
[tex]\[ V = 25000 \times (1 - 0.20)^{4.5} \][/tex]
3. Calculate the depreciation factor:
[tex]\[ 1 - 0.20 = 0.80 \][/tex]
4. Raise the depreciation factor to the power of [tex]\( t = 4.5 \)[/tex]:
[tex]\[ 0.80^{4.5} \approx 0.36635708 \][/tex]
5. Multiply the initial price by this result:
[tex]\[ V = 25000 \times 0.36635708 \][/tex]
[tex]\[ V \approx 9158.934 \][/tex]
Thus, the exact value of the car after [tex]\( 4.5 \)[/tex] years is approximately [tex]\( \$9158.934 \)[/tex].
6. Round the value to the nearest available option:
The available options are:
- [tex]\( \$2500 \)[/tex]
- [tex]\( \$9159 \)[/tex]
- [tex]\( \$22827 \)[/tex]
- [tex]\( \$23791 \)[/tex]
Inspecting these options, the closest value to [tex]\( \$9158.934 \)[/tex] is [tex]\( \$9159 \)[/tex].
Therefore, the approximate value of Doug's car after [tex]\( 4.5 \)[/tex] years is
[tex]\[ \boxed{\$9159} \][/tex]
