To determine when the car will be worth $4,000 using the depreciation function \( y = 21,000(1 - 0.15)^x \), you can set the equation equal to $4,000 and solve for \( x \):
\[ 21,000(1 - 0.15)^x = 4,000. \]
First, simplify the equation:
\[ (1 - 0.15)^x = \frac{4,000}{21,000}, \]
which simplifies to:
\[ (0.85)^x = \frac{4}{21} \approx 0.1905. \]
Next, take the logarithm of both sides:
\[ \log((0.85)^x) = \log(0.1905). \]
Using the power rule of logarithms:
\[ x \cdot \log(0.85) = \log(0.1905). \]
Now, solve for \( x \):
\[ x = \frac{\log(0.1905)}{\log(0.85)}. \]
Calculate the values:
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Calculate \( \log(0.1905) \) and \( \log(0.85) \) using a calculator:
- \( \log(0.1905) \approx -0.719 \)
- \( \log(0.85) \approx -0.0682 \)
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Now plug these values into the formula: \[ x \approx \frac{-0.719}{-0.0682} \approx 10.54. \]
From the result, we see that after approximately 10.5 years, the car will be worth $4,000.
Conclusion
The best statement to describe when the car will be worth $4,000 is:
Statement #1: The car will have a value of $4,000 after roughly 10.2 years.
(This statement is closer to the calculated value of 10.5 years compared to Statement #2.)
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