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Doug bought a new car for [tex]\$25,000[/tex]. He estimates his car will depreciate, or lose value, at a rate of [tex]20\%[/tex] per year. The value of his car is modeled by the equation [tex]V = P(1-r)^t[/tex], where [tex]V[/tex] is the value of the car, [tex]P[/tex] is the price he paid, [tex]r[/tex] is the annual rate of depreciation, and [tex]t[/tex] is the number of years he has owned the car.

According to the model, what will be the approximate value of his car after [tex]4 \frac{1}{2}[/tex] years?

A. [tex]\$2,500[/tex]
B. [tex]\$9,159[/tex]
C. [tex]\$22,827[/tex]
D. [tex]\$23,791[/tex]



Answer :

GinnyAnswer
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]

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