To determine when the motorcycle will be worth $7,000 based on its depreciation function, we can set the equation for the motorcycle's value equal to $7,000 and solve for \( x \):
The motorcycle's value function is given by:
\[ y = 14,000(1 - 0.09)^x \]
Setting \( y \) to $7,000:
\[ 7,000 = 14,000(1 - 0.09)^x \]
Dividing both sides by 14,000:
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
This simplifies to:
\[ 0.5 = (0.91)^x \]
Now we take the logarithm of both sides to solve for \( x \):
\[ \log(0.5) = x \cdot \log(0.91) \]
Now we solve for \( x \):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator or logarithm table, find the values of \( \log(0.5) \) and \( \log(0.91) \):
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now substituting these into the equation:
\[ x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
This means that the motorcycle will have a value of $7,000 after roughly 6.93 years.
Given the provided statements:
- Statement #1: The motorcycle will have a value of $7,000 after roughly 7.3 years.
- Statement #2: The motorcycle will have a value of $7,000 after roughly 6.7 years.
Since the calculated value is approximately 6.93, neither statement is exactly correct, but Statement #2 ("roughly 6.7 years") is closer to the calculated value. Thus, if you have to choose the best option between the two, you should select Statement #2.
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