Question
Salim purchased a new car for $21,000, and it depreciates in value each year. The function y=21,000(1−0.15)x represents the depreciation. He also purchases a motorcycle, which depreciates each year as well. Its depreciation is represented by the function y=14,000(1−0.09)x. Graph these two functions using GeoGebra. Which statement best describes when the motorcycle will be worth $7,000? 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. (1 point) Statement #
Answers
GPT-4o mini
To determine when the motorcycle will be worth $7,000, we can solve the depreciation function for the motorcycle given by:
\[ y = 14,000(1 - 0.09)^x \]
We need to find \( x \) such that:
\[ 14,000(1 - 0.09)^x = 7,000 \]
Dividing both sides by 14,000:
\[ (1 - 0.09)^x = \frac{7,000}{14,000} \]
This simplifies to:
\[ (0.91)^x = 0.5 \]
To solve for \( x \), we can use logarithms:
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Calculating the values:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now divide:
\[ x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
So, the motorcycle will have a value of $7,000 after approximately 6.93 years. This result indicates that the motorcycle will be worth $7,000 after roughly **6.7 years**, making **Statement #2** the better description.
Thus, **Statement #2**: The motorcycle will have a value of $7,000 after roughly 6.7 years is the correct answer.
\[ y = 14,000(1 - 0.09)^x \]
We need to find \( x \) such that:
\[ 14,000(1 - 0.09)^x = 7,000 \]
Dividing both sides by 14,000:
\[ (1 - 0.09)^x = \frac{7,000}{14,000} \]
This simplifies to:
\[ (0.91)^x = 0.5 \]
To solve for \( x \), we can use logarithms:
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Calculating the values:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now divide:
\[ x \approx \frac{-0.3010}{-0.0434} \approx 6.93 \]
So, the motorcycle will have a value of $7,000 after approximately 6.93 years. This result indicates that the motorcycle will be worth $7,000 after roughly **6.7 years**, making **Statement #2** the better description.
Thus, **Statement #2**: The motorcycle will have a value of $7,000 after roughly 6.7 years is the correct answer.