Question
Salim purchases 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 car will be worth $4,000?
Statement #1: The car will have a value of $4,000 after roughly 10.2 years.
Statement #2: The car will have a value of $4,000 after roughly 13.3 years.
(1 point)
Statement #
Statement #1: The car will have a value of $4,000 after roughly 10.2 years.
Statement #2: The car will have a value of $4,000 after roughly 13.3 years.
(1 point)
Statement #
Answers
GPT-4o mini
To determine which statement is correct regarding when the car will be worth $4,000, we need to solve the equation \(21,000(1 - 0.15)^x = 4,000\).
1. First, we simplify the equation:
\[
21,000(0.85)^x = 4,000
\]
2. Divide both sides by 21,000:
\[
(0.85)^x = \frac{4,000}{21,000}
\]
Simplifying the right side gives:
\[
(0.85)^x = \frac{4}{21} \approx 0.1905
\]
3. Now, take the logarithm of both sides. We can use natural logarithm (ln):
\[
\ln((0.85)^x) = \ln(0.1905)
\]
This simplifies to:
\[
x \cdot \ln(0.85) = \ln(0.1905)
\]
4. Solve for \(x\) by dividing both sides by \(\ln(0.85)\):
\[
x = \frac{\ln(0.1905)}{\ln(0.85)}
\]
5. Calculate the values using a calculator:
- \(\ln(0.1905) \approx -1.657\)
- \(\ln(0.85) \approx -0.1625\)
6. Thus,
\[
x \approx \frac{-1.657}{-0.1625} \approx 10.2
\]
From the calculation, the car will have a value of $4,000 after roughly **10.2 years**. Therefore, the correct statement is:
**Statement #1: The car will have a value of $4,000 after roughly 10.2 years.**
1. First, we simplify the equation:
\[
21,000(0.85)^x = 4,000
\]
2. Divide both sides by 21,000:
\[
(0.85)^x = \frac{4,000}{21,000}
\]
Simplifying the right side gives:
\[
(0.85)^x = \frac{4}{21} \approx 0.1905
\]
3. Now, take the logarithm of both sides. We can use natural logarithm (ln):
\[
\ln((0.85)^x) = \ln(0.1905)
\]
This simplifies to:
\[
x \cdot \ln(0.85) = \ln(0.1905)
\]
4. Solve for \(x\) by dividing both sides by \(\ln(0.85)\):
\[
x = \frac{\ln(0.1905)}{\ln(0.85)}
\]
5. Calculate the values using a calculator:
- \(\ln(0.1905) \approx -1.657\)
- \(\ln(0.85) \approx -0.1625\)
6. Thus,
\[
x \approx \frac{-1.657}{-0.1625} \approx 10.2
\]
From the calculation, the car will have a value of $4,000 after roughly **10.2 years**. Therefore, the correct statement is:
**Statement #1: The car will have a value of $4,000 after roughly 10.2 years.**