To determine when the motorcycle will be worth $7,000, you can set the depreciation function for the motorcycle equal to $7,000 and solve for \(x\):
The function for the motorcycle is given by: \[ y = 14,000(1 - 0.09)^x \]
Set \(y\) equal to $7,000: \[ 14,000(1 - 0.09)^x = 7,000 \]
Now, simplify and solve for \(x\): \[ (1 - 0.09)^x = \frac{7,000}{14,000} \] \[ (0.91)^x = 0.5 \]
Next, take the logarithm of both sides: \[ \log((0.91)^x) = \log(0.5) \]
Using the property of logarithms: \[ x \cdot \log(0.91) = \log(0.5) \] \[ x = \frac{\log(0.5)}{\log(0.91)} \]
Now, calculate the value:
- \(\log(0.5) \approx -0.3010\)
- \(\log(0.91) \approx -0.0434\)
Thus, \[ x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
This value is approximately 6.9 years, which is close to 6.7 years but not quite equal.
Given the options:
- 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.
The closest statement to our calculation is Statement #2, which suggests that the motorcycle will be worth $7,000 after roughly 6.7 years.