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 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
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
Statement #_________
Answers
GPT-4o mini
To determine when the car will be worth $4,000 according to the depreciation function \( y = 21,000(1 - 0.15)^x \), we want to find the value of \( x \) at which \( y = 4,000 \).
1. Set the equation equal to \( 4,000 \):
\[
4,000 = 21,000(1 - 0.15)^x
\]
2. Divide both sides by \( 21,000 \):
\[
\frac{4,000}{21,000} = (1 - 0.15)^x
\]
\[
\frac{4}{21} \approx 0.1905 \quad \text{(Calculating this gives approximately 0.1905)}
\]
3. Taking the natural logarithm on both sides:
\[
\ln\left(\frac{4}{21}\right) = \ln\left((0.85)^x\right)
\]
4. Apply the power rule for logarithms:
\[
\ln\left(\frac{4}{21}\right) = x \cdot \ln(0.85)
\]
5. Solving for \( x \):
\[
x = \frac{\ln\left(\frac{4}{21}\right)}{\ln(0.85)}
\]
Now, calculating the value gives:
- \( \ln\left(\frac{4}{21}\right) \approx -1.4816 \)
- \( \ln(0.85) \approx -0.1625 \)
Thus:
\[
x \approx \frac{-1.4816}{-0.1625} \approx 9.1 \text{ years (approximately)}
\]
The statement seems to indicate that the car will have a value of $4,000, which gives us the insight that neither Statement #1 nor Statement #2 exactly matches this outcome since neither addresses 9.1 years directly.
Thus, you should make a decision based on your calculations. Since the years provided are around 10.2 and 13.3, both are incorrect. Therefore, the most accurate conclusion we can derive from this calculation is that neither statement fully captures the accurate timeline for when the car will reach a value of $4,000.
**Final Answer: None of the statements accurately describe when the car will be worth $4,000.**
1. Set the equation equal to \( 4,000 \):
\[
4,000 = 21,000(1 - 0.15)^x
\]
2. Divide both sides by \( 21,000 \):
\[
\frac{4,000}{21,000} = (1 - 0.15)^x
\]
\[
\frac{4}{21} \approx 0.1905 \quad \text{(Calculating this gives approximately 0.1905)}
\]
3. Taking the natural logarithm on both sides:
\[
\ln\left(\frac{4}{21}\right) = \ln\left((0.85)^x\right)
\]
4. Apply the power rule for logarithms:
\[
\ln\left(\frac{4}{21}\right) = x \cdot \ln(0.85)
\]
5. Solving for \( x \):
\[
x = \frac{\ln\left(\frac{4}{21}\right)}{\ln(0.85)}
\]
Now, calculating the value gives:
- \( \ln\left(\frac{4}{21}\right) \approx -1.4816 \)
- \( \ln(0.85) \approx -0.1625 \)
Thus:
\[
x \approx \frac{-1.4816}{-0.1625} \approx 9.1 \text{ years (approximately)}
\]
The statement seems to indicate that the car will have a value of $4,000, which gives us the insight that neither Statement #1 nor Statement #2 exactly matches this outcome since neither addresses 9.1 years directly.
Thus, you should make a decision based on your calculations. Since the years provided are around 10.2 and 13.3, both are incorrect. Therefore, the most accurate conclusion we can derive from this calculation is that neither statement fully captures the accurate timeline for when the car will reach a value of $4,000.
**Final Answer: None of the statements accurately describe when the car will be worth $4,000.**