To determine when the motorcycle will be worth $7,000, we need to solve the equation given by the depreciation function for the motorcycle:
The depreciation function for the motorcycle is given by: \[ y = 14,000(1 - 0.09)^x \]
We need to set \( y \) to $7,000 and solve for \( x \):
\[ 7,000 = 14,000(1 - 0.09)^x \]
First, we can simplify this equation:
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
This simplifies to:
\[ 0.5 = (0.91)^x \]
Next, we can take the logarithm of both sides to solve for \( x \):
\[ \log(0.5) = x \cdot \log(0.91) \]
Now, we isolate \( x \):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator to find the logarithmic values:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Therefore:
\[ x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
This means the motorcycle will be worth $7,000 after approximately 6.93 years.
Given the choices:
- 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 best statement that aligns with our finding (6.93 years) is Statement #2.
Thus, the answer is Statement #2.