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.
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.
Answers
GPT-4o mini
To analyze when the motorcycle will be worth $7,000, we can start by using the depreciation function provided:
\[
y = 14,000(1 - 0.09)^x
\]
We want to find the value of \( x \) when \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
Dividing both sides by 14,000:
\[
\frac{7,000}{14,000} = (1 - 0.09)^x
\]
Simplifying the left side:
\[
0.5 = (0.91)^x
\]
Next, we will use logarithms to solve for \( x \):
\[
\log(0.5) = x \cdot \log(0.91)
\]
Solving for \( x \):
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Using a calculator or logarithm tables to find the values:
- \(\log(0.5) \approx -0.3010\)
- \(\log(0.91) \approx -0.0434\)
Now compute \( x \):
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
Rounding this, we find that \( x \approx 6.9 \) years.
Now, looking at the statements provided:
- 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 we calculated that the motorcycle will be worth $7,000 after approximately 6.9 years, Statement #2 is the closest estimate, although it is slightly off.
Therefore, **Statement #2** is more accurate.
\[
y = 14,000(1 - 0.09)^x
\]
We want to find the value of \( x \) when \( y = 7,000 \):
\[
7,000 = 14,000(1 - 0.09)^x
\]
Dividing both sides by 14,000:
\[
\frac{7,000}{14,000} = (1 - 0.09)^x
\]
Simplifying the left side:
\[
0.5 = (0.91)^x
\]
Next, we will use logarithms to solve for \( x \):
\[
\log(0.5) = x \cdot \log(0.91)
\]
Solving for \( x \):
\[
x = \frac{\log(0.5)}{\log(0.91)}
\]
Using a calculator or logarithm tables to find the values:
- \(\log(0.5) \approx -0.3010\)
- \(\log(0.91) \approx -0.0434\)
Now compute \( x \):
\[
x \approx \frac{-0.3010}{-0.0434} \approx 6.93
\]
Rounding this, we find that \( x \approx 6.9 \) years.
Now, looking at the statements provided:
- 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 we calculated that the motorcycle will be worth $7,000 after approximately 6.9 years, Statement #2 is the closest estimate, although it is slightly off.
Therefore, **Statement #2** is more accurate.