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To determine the value of [tex]\( P \)[/tex], which is the initial price at which Jason bought the car in 2020, given that the value of the car in 2022 is \[tex]$12,696, let's analyze how the depreciation works over the years.
1. Identify the given values and unknowns:
- The car’s value in 2022, which is \( t = 2 \) years after 2020, is given as \$[/tex]12,696.
- We are given four possible values for [tex]\( P \)[/tex] (the initial price in 2020): [tex]\(\$ 6,900\)[/tex], [tex]\(\$ 13,800\)[/tex], [tex]\(\$ 15,000\)[/tex], and [tex]\(\$ 20,000\)[/tex].
- Let [tex]\( r \)[/tex] be the annual depreciation rate.
2. Set up the depreciation formula:
- The value of the car decreases at a fixed percentage rate each year. If the initial value in 2020 is [tex]\( P \)[/tex], then the value after [tex]\( t \)[/tex] years can be modeled as:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
- Given [tex]\( t = 2 \)[/tex], we can write:
[tex]\[ V(2) = P \times (1 - r)^2 \][/tex]
- Substituting [tex]\( V(2) = 12,696 \)[/tex], we get:
[tex]\[ 12,696 = P \times (1 - r)^2 \][/tex]
3. Solve for each possible option of [tex]\( P \)[/tex]:
- Option (A) [tex]\( P = 6,900 \)[/tex]:
[tex]\[ 12,696 = 6,900 \times (1 - r)^2 \quad (\text{which clearly does not fit as } 12,696 > 6,900) \][/tex]
- Option (B) [tex]\( P = 13,800 \)[/tex]:
[tex]\[ 12,696 = 13,800 \times (1 - r)^2 \][/tex]
Dividing both sides by 13,800,
[tex]\[ \frac{12,696}{13,800} = (1 - r)^2 \][/tex]
Simplifying,
[tex]\[ 1 - r = \sqrt{\frac{12,696}{13,800}} \][/tex]
Solving the square root,
[tex]\[ 1 - r = \sqrt{0.92} \][/tex]
Therefore,
[tex]\[ r = 1 - \sqrt{0.92} \][/tex]
- Option (C) [tex]\( P = 15,000 \)[/tex]:
[tex]\[ 12,696 \ne 15,000 \times (1 - r)^2 \quad (\text{as } 12,696 < 15,000) \][/tex]
- Option (D) [tex]\( P = 20,000 \)[/tex]:
[tex]\[ 12,696 \ne 20,000 \times (1 - r)^2 \quad (\text{as } 12,696 < 20,000) \][/tex]
4. Validating Option (B):
- Calculating [tex]\( \sqrt{\frac{12,696}{13,800}} \)[/tex]:
[tex]\[ \sqrt{0.92} \approx 0.96 \][/tex]
Thus, the calculated depreciation rate [tex]\( r \)[/tex] is:
[tex]\[ r = 1 - 0.96 \approx 0.04 \][/tex]
Therefore, the initial price of the car [tex]\( P \)[/tex] is [tex]\(\$ 13,800\)[/tex] and the depreciation rate is approximately [tex]\( 1 - \frac{\sqrt{23}}{5} \)[/tex], confirming that:
- The value of [tex]\( P \)[/tex] is [tex]\(\$ 13,800 \)[/tex].
So, the correct answer is:
(B) [tex]\( P = \$13,800 \)[/tex]
- We are given four possible values for [tex]\( P \)[/tex] (the initial price in 2020): [tex]\(\$ 6,900\)[/tex], [tex]\(\$ 13,800\)[/tex], [tex]\(\$ 15,000\)[/tex], and [tex]\(\$ 20,000\)[/tex].
- Let [tex]\( r \)[/tex] be the annual depreciation rate.
2. Set up the depreciation formula:
- The value of the car decreases at a fixed percentage rate each year. If the initial value in 2020 is [tex]\( P \)[/tex], then the value after [tex]\( t \)[/tex] years can be modeled as:
[tex]\[ V(t) = P \times (1 - r)^t \][/tex]
- Given [tex]\( t = 2 \)[/tex], we can write:
[tex]\[ V(2) = P \times (1 - r)^2 \][/tex]
- Substituting [tex]\( V(2) = 12,696 \)[/tex], we get:
[tex]\[ 12,696 = P \times (1 - r)^2 \][/tex]
3. Solve for each possible option of [tex]\( P \)[/tex]:
- Option (A) [tex]\( P = 6,900 \)[/tex]:
[tex]\[ 12,696 = 6,900 \times (1 - r)^2 \quad (\text{which clearly does not fit as } 12,696 > 6,900) \][/tex]
- Option (B) [tex]\( P = 13,800 \)[/tex]:
[tex]\[ 12,696 = 13,800 \times (1 - r)^2 \][/tex]
Dividing both sides by 13,800,
[tex]\[ \frac{12,696}{13,800} = (1 - r)^2 \][/tex]
Simplifying,
[tex]\[ 1 - r = \sqrt{\frac{12,696}{13,800}} \][/tex]
Solving the square root,
[tex]\[ 1 - r = \sqrt{0.92} \][/tex]
Therefore,
[tex]\[ r = 1 - \sqrt{0.92} \][/tex]
- Option (C) [tex]\( P = 15,000 \)[/tex]:
[tex]\[ 12,696 \ne 15,000 \times (1 - r)^2 \quad (\text{as } 12,696 < 15,000) \][/tex]
- Option (D) [tex]\( P = 20,000 \)[/tex]:
[tex]\[ 12,696 \ne 20,000 \times (1 - r)^2 \quad (\text{as } 12,696 < 20,000) \][/tex]
4. Validating Option (B):
- Calculating [tex]\( \sqrt{\frac{12,696}{13,800}} \)[/tex]:
[tex]\[ \sqrt{0.92} \approx 0.96 \][/tex]
Thus, the calculated depreciation rate [tex]\( r \)[/tex] is:
[tex]\[ r = 1 - 0.96 \approx 0.04 \][/tex]
Therefore, the initial price of the car [tex]\( P \)[/tex] is [tex]\(\$ 13,800\)[/tex] and the depreciation rate is approximately [tex]\( 1 - \frac{\sqrt{23}}{5} \)[/tex], confirming that:
- The value of [tex]\( P \)[/tex] is [tex]\(\$ 13,800 \)[/tex].
So, the correct answer is:
(B) [tex]\( P = \$13,800 \)[/tex]
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