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 Desmos. 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 Desmos. 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 set the depreciation function for the motorcycle equal to $7,000 and solve for \( x \).
The equation for the motorcycle is:
\[
y = 14,000(1 - 0.09)^x
\]
Setting \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
Dividing both sides by 14,000:
\[
\frac{7,000}{14,000} = (0.91)^x
\]
This simplifies to:
\[
0.5 = (0.91)^x
\]
Next, we can take the logarithm of both sides:
\[
\log(0.5) = x \cdot \log(0.91)
\]
Now, solving for \( x \):
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Calculating the values:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now substituting these values:
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
This result means that the motorcycle will be worth $7,000 after approximately **6.9 years**.
Based on the options provided:
- **Statement #1:** The motorcycle will have a value of $7,000 after roughly 7.3 years. (Not correct)
- **Statement #2:** The motorcycle will have a value of $7,000 after roughly 6.7 years. (Closer to the calculated value)
Though neither statement is exact, Statement #2 is the best choice as it is closer to the calculated depreciation time.
So the correct choice is:
**Statement #2:** The motorcycle will have a value of $7,000 after roughly 6.7 years.
The equation for the motorcycle is:
\[
y = 14,000(1 - 0.09)^x
\]
Setting \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
Dividing both sides by 14,000:
\[
\frac{7,000}{14,000} = (0.91)^x
\]
This simplifies to:
\[
0.5 = (0.91)^x
\]
Next, we can take the logarithm of both sides:
\[
\log(0.5) = x \cdot \log(0.91)
\]
Now, solving for \( x \):
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Calculating the values:
- \( \log(0.5) \approx -0.3010 \)
- \( \log(0.91) \approx -0.0434 \)
Now substituting these values:
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
This result means that the motorcycle will be worth $7,000 after approximately **6.9 years**.
Based on the options provided:
- **Statement #1:** The motorcycle will have a value of $7,000 after roughly 7.3 years. (Not correct)
- **Statement #2:** The motorcycle will have a value of $7,000 after roughly 6.7 years. (Closer to the calculated value)
Though neither statement is exact, Statement #2 is the best choice as it is closer to the calculated depreciation time.
So the correct choice is:
**Statement #2:** The motorcycle will have a value of $7,000 after roughly 6.7 years.