To determine when the car will be worth $4,000 based on the depreciation function \( y = 21,000(1 - 0.15)^x \), we need to set \( y \) to $4,000 and solve for \( x \):
\[ 4000 = 21000(1 - 0.15)^x \]
Dividing both sides by 21,000:
\[ \frac{4000}{21000} = (0.85)^x \]
\[ \frac{4}{21} \approx 0.1905 = (0.85)^x \]
Now we apply the logarithm to both sides to solve for \( x \):
\[ \log\left(\frac{4}{21}\right) = x \cdot \log(0.85) \]
Solving for \( x \):
\[ x = \frac{\log\left(\frac{4}{21}\right)}{\log(0.85)} \]
Now, performing the calculations:
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Calculate \( \log\left(\frac{4}{21}\right) \):
- \( \frac{4}{21} \approx 0.1905 \)
- \( \log(0.1905) \approx -0.720 \)
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Calculate \( \log(0.85) \):
- \( \log(0.85) \approx -0.069 \)
Now plug these into the equation for \( x \):
\[ x \approx \frac{-0.720}{-0.069} \approx 10.43 \]
Thus, the car will be worth $4,000 after approximately 10.43 years.
Comparing our result:
- Statement #1: The car will have a value of $4,000 after roughly 10.2 years.
- Statement #2: The car will have a value of $4,000 after roughly 13.3 years.
The correct choice is Statement #1, since it is closer to our calculated value of approximately 10.43 years.