To find out when the motorcycle will be worth $7000, we need to solve the equation given by the depreciation function:
\[ f(x) = 14000(1−0.09)^x = 7000 \]
First, we can simplify the equation:
\[ 14000(1−0.09)^x = 7000 \]
Dividing both sides by 14000 gives:
\[ (1−0.09)^x = \frac{7000}{14000} \]
Simplifying the right side:
\[ (1−0.09)^x = 0.5 \]
Now substituting \(1 - 0.09\) with \(0.91\):
\[ 0.91^x = 0.5 \]
To solve for \(x\), we take the logarithm of both sides:
\[ \log(0.91^x) = \log(0.5) \]
Using the power rule of logarithms:
\[ x \cdot \log(0.91) = \log(0.5) \]
Now, divide both sides by \(\log(0.91)\):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Now, we can calculate the values using a calculator:
- \( \log(0.5) \approx -0.3010\)
- \( \log(0.91) \approx -0.0435\)
Substituting these values in:
\[ x \approx \frac{-0.3010}{-0.0435} \approx 6.92 \]
Thus, it will take approximately 6.9 years for the motorcycle to be worth $7000. Given the options, the closest is 7 years.
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