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 Demos. 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.
Statement # _________
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.
Statement # _________
Answers
GPT-4o mini
To determine when the motorcycle will be worth $7,000, we can use the depreciation function for the motorcycle, which is given by:
\[ y = 14,000(1 - 0.09)^x \]
We want to find \( x \) when \( y = 7,000 \). Setting up the equation:
\[ 7,000 = 14,000(1 - 0.09)^x \]
Dividing both sides by \( 14,000 \):
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
This simplifies to:
\[ 0.5 = (0.91)^x \]
Next, we need to solve for \( x \) by taking the logarithm of both sides:
\[ \log(0.5) = x \cdot \log(0.91) \]
Now, isolating \( x \):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator to find the logarithms:
\[
\log(0.5) \approx -0.3010
\]
\[
\log(0.91) \approx -0.0434
\]
So:
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
This result indicates that the motorcycle will be worth $7,000 after approximately 6.93 years.
Comparing this with the provided 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.
Based on our calculation of approximately 6.93 years, the motorcycle will reach a value of $7,000 after roughly **6.9 years**. Therefore, the more accurate choice is **Statement #2**.
Thus, the answer is:
**Statement #2**.
\[ y = 14,000(1 - 0.09)^x \]
We want to find \( x \) when \( y = 7,000 \). Setting up the equation:
\[ 7,000 = 14,000(1 - 0.09)^x \]
Dividing both sides by \( 14,000 \):
\[ \frac{7,000}{14,000} = (1 - 0.09)^x \]
This simplifies to:
\[ 0.5 = (0.91)^x \]
Next, we need to solve for \( x \) by taking the logarithm of both sides:
\[ \log(0.5) = x \cdot \log(0.91) \]
Now, isolating \( x \):
\[ x = \frac{\log(0.5)}{\log(0.91)} \]
Using a calculator to find the logarithms:
\[
\log(0.5) \approx -0.3010
\]
\[
\log(0.91) \approx -0.0434
\]
So:
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
This result indicates that the motorcycle will be worth $7,000 after approximately 6.93 years.
Comparing this with the provided 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.
Based on our calculation of approximately 6.93 years, the motorcycle will reach a value of $7,000 after roughly **6.9 years**. Therefore, the more accurate choice is **Statement #2**.
Thus, the answer is:
**Statement #2**.