To find out when the motorcycle will be worth $7000, we need to solve the equation given by the depreciation function for \( f(x) = 14000(1 - 0.09)^x \), where \( f(x) \) represents the value of the motorcycle after \( x \) years.
We can set this equal to $7000:
\[ 14000(1 - 0.09)^x = 7000 \]
First, divide both sides by 14000:
\[ (1 - 0.09)^x = \frac{7000}{14000} \]
This simplifies to:
\[ (1 - 0.09)^x = 0.5 \]
Next, calculate \( 1 - 0.09 \):
\[ 1 - 0.09 = 0.91 \]
So the equation becomes:
\[ 0.91^x = 0.5 \]
Now we can solve for \( x \) by taking the logarithm of both sides. Using the natural logarithm (you can also use any logarithm):
\[ \ln(0.91^x) = \ln(0.5) \]
Using the property of logarithms:
\[ x \ln(0.91) = \ln(0.5) \]
Now, we can solve for \( x \):
\[ x = \frac{\ln(0.5)}{\ln(0.91)} \]
Calculating the right side:
- \( \ln(0.5) \approx -0.6931 \)
- \( \ln(0.91) \approx -0.0943 \)
Now substitute these values:
\[ x \approx \frac{-0.6931}{-0.0943} \approx 7.35 \]
Thus, \( x \approx 7.3 \) years.
The motorcycle will be worth $7000 approximately after 7.3 years.
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