To analyze the situation described, let's assume that the value of the motorcycle depreciates according to the function given. The equation seems to represent a linear depreciation model, where a motorcycle starts with a value of $14,000 and depreciates by a rate of 9% per year.
The value \( V(t) \) of the motorcycle after \( t \) years can be expressed as: \[ V(t) = 14000(1 - 0.09)^t \]
This simplifies our function: \[ V(t) = 14000(0.91)^t \]
To find out when the motorcycle will be worth $7,000, we can set \( V(t) \) equal to $7,000 and solve for \( t \):
\[ 7000 = 14000(0.91)^t \]
Divide both sides by 14,000: \[ 0.5 = (0.91)^t \]
To solve for \( t \), we can take the logarithm of both sides: \[ \log(0.5) = t \cdot \log(0.91) \]
Now, solve for \( t \): \[ t = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now substituting those values into the equation for \( t \): \[ t \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
So, the motorcycle will be worth $7,000 in approximately 6.93 years.
To address your question about whether the motorcycle will be worth $7,000 in 7.3 years or 6.7 years, we can determine that it would be closer to 7.0 years. Thus, the correct statement would be that the motorcycle will be worth $7,000 before 7.3 years and after 6.7 years.
In summary, the motorcycle will be worth $7,000 approximately at 6.93 years, which is closer to 7.3 years than 6.7 years.