To find out when the motorcycle will be worth $7000, we need to set the given depreciation function equal to $7000 and solve for \( x \):
\[ f(x) = 14000(1 - 0.09)^x \]
Setting this equal to $7000:
\[ 14000(1 - 0.09)^x = 7000 \]
Next, divide both sides by 14000:
\[ (1 - 0.09)^x = \frac{7000}{14000} \]
This simplifies to:
\[ (1 - 0.09)^x = 0.5 \]
Substituting \( 1 - 0.09 \) with \( 0.91 \):
\[ 0.91^x = 0.5 \]
To solve for \( x \), we can take the logarithm of both sides. Using the natural logarithm:
\[ \ln(0.91^x) = \ln(0.5) \]
Using the power rule of logarithms:
\[ x \cdot \ln(0.91) = \ln(0.5) \]
Now, solve for \( x \):
\[ x = \frac{\ln(0.5)}{\ln(0.91)} \]
Calculating the values:
- \( \ln(0.5) \approx -0.6931 \)
- \( \ln(0.91) \approx -0.0943 \)
So,
\[ x \approx \frac{-0.6931}{-0.0943} \approx 7.35 \]
Since \( x \) represents years, we round to the nearest whole number. This means the motorcycle will be worth approximately $7000 after about 7 years.
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