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 #
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 analyze its depreciation function, which is given by:
\[ y = 14,000(1 - 0.09)^x \]
We need to find \( x \) when \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
This can be simplified to:
\[
(1 - 0.09)^x = \frac{7,000}{14,000}
\]
\[
(0.91)^x = 0.5
\]
Now, we apply the logarithm to both sides to solve for \( x \):
\[
\log((0.91)^x) = \log(0.5)
\]
\[
x \cdot \log(0.91) = \log(0.5)
\]
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Now, we can calculate this using a calculator:
1. Calculate \( \log(0.5) \):
- Approximately \(-0.3010\)
2. Calculate \( \log(0.91) \):
- Approximately \(-0.0434\)
3. Now find \( x \):
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
Based on this calculation, we can say that the motorcycle will have a value of $7,000 after approximately 6.93 years.
Given 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.
The closest statement is **Statement #2**, since 6.93 is closer to 6.7 than it is to 7.3.
Therefore, the answer is:
**Statement #2**.
\[ y = 14,000(1 - 0.09)^x \]
We need to find \( x \) when \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
This can be simplified to:
\[
(1 - 0.09)^x = \frac{7,000}{14,000}
\]
\[
(0.91)^x = 0.5
\]
Now, we apply the logarithm to both sides to solve for \( x \):
\[
\log((0.91)^x) = \log(0.5)
\]
\[
x \cdot \log(0.91) = \log(0.5)
\]
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Now, we can calculate this using a calculator:
1. Calculate \( \log(0.5) \):
- Approximately \(-0.3010\)
2. Calculate \( \log(0.91) \):
- Approximately \(-0.0434\)
3. Now find \( x \):
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
Based on this calculation, we can say that the motorcycle will have a value of $7,000 after approximately 6.93 years.
Given 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.
The closest statement is **Statement #2**, since 6.93 is closer to 6.7 than it is to 7.3.
Therefore, the answer is:
**Statement #2**.