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.
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.
Answers
GPT-4o mini
To determine when the motorcycle will be worth $7,000, we need to analyze the depreciation function provided for the motorcycle:
The function for the motorcycle is:
\[ y = 14,000(1 - 0.09)^x \]
We want to find \( x \) when \( y = 7,000 \):
\[ 7,000 = 14,000(1 - 0.09)^x \]
Now, we can simplify the equation:
1. Divide both sides by 14,000:
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
\[ 0.5 = (0.91)^x \]
2. Next, we can solve for \( x \) using logarithms:
\[ x = \log_{0.91}(0.5) \]
We can rewrite this using the change of base formula:
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Now we can calculate the values using a calculator:
1. Calculate \( \log(0.5) \):
- \( \log(0.5) \approx -0.3010 \)
2. Calculate \( \log(0.91) \):
- \( \log(0.91) \approx -0.0434 \)
3. 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, which is roughly 7 years.
Looking at the statements:
- **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.
Since our calculation shows that it is closer to 7 years, **Statement #2** is the more accurate statement, even if it might not match exactly. **Statement #1** significantly overestimates the time it takes to reach a $7,000 value.
Thus:
**The motorcycle will have a value of $7,000 after roughly 6.7 years (Statement #2) is the best choice.**
The function for the motorcycle is:
\[ y = 14,000(1 - 0.09)^x \]
We want to find \( x \) when \( y = 7,000 \):
\[ 7,000 = 14,000(1 - 0.09)^x \]
Now, we can simplify the equation:
1. Divide both sides by 14,000:
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
\[ 0.5 = (0.91)^x \]
2. Next, we can solve for \( x \) using logarithms:
\[ x = \log_{0.91}(0.5) \]
We can rewrite this using the change of base formula:
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Now we can calculate the values using a calculator:
1. Calculate \( \log(0.5) \):
- \( \log(0.5) \approx -0.3010 \)
2. Calculate \( \log(0.91) \):
- \( \log(0.91) \approx -0.0434 \)
3. 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, which is roughly 7 years.
Looking at the statements:
- **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.
Since our calculation shows that it is closer to 7 years, **Statement #2** is the more accurate statement, even if it might not match exactly. **Statement #1** significantly overestimates the time it takes to reach a $7,000 value.
Thus:
**The motorcycle will have a value of $7,000 after roughly 6.7 years (Statement #2) is the best choice.**