The lapse rate is the rate at which the temperature decreases in the atmosphere with respect to increasing altitude. Concurrent measurements indicate that at an elevation of 6.6 km, the temperature is - 8.4°C, and at an elevation of 3.5 km, the temperature is 9.6°C. Based on the Mean Value Theorem, can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation? Explain. Can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation? O A. Yes. At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value. OB. No. At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value. O C. Yes. At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value. O D. No. At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
The lapse rate is the rate at which the temperature decreases in the atmosphere with respect to increasing altitude. Concurrent measurements indicate that at an elevation of 6.6 km, the temperature is - 8.4°C, and at an elevation of 3.5 km, the temperature is 9.6°C. Based on the Mean Value Theorem, can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation? Explain. Can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation? O A. Yes. At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value. OB. No. At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value. O C. Yes. At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value. O D. No. At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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![**Understanding the Lapse Rate in Meteorology: An Application of the Mean Value Theorem**
The lapse rate is the rate at which the temperature decreases in the atmosphere with respect to increasing altitude. Concurrent measurements indicate that at an elevation of 6.6 km, the temperature is -8.4°C, and at an elevation of 3.5 km, the temperature is 9.6°C. Based on the Mean Value Theorem, can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation?
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### Analysis Question:
**Can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation?**
---
**Answer Choices:**
- **A. Yes.** At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value.
- **B. No.** At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value.
- **C. Yes.** At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
- **D. No.** At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
---
### Explanation:
To determine whether the lapse rate exceeds the threshold value of 3°C/km at any point between the given altitudes, we apply the Mean Value Theorem. This theorem states that if a function is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one \(c\) in the interval \((a, b)\) such that the derivative at \(c\) equals the average rate of change of the function over \([a, b]\).
Let's calculate the average rate of change (lapse rate) between the altitudes 3.5 km and 6.6 km:
1. **Temperature at 6.6 km:** -8.4°C
2. **Temperature at 3.5 km:** 9.6°C
3. **Change in temperature](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F66e01285-6633-4a96-9df0-a35da9560d14%2F0b94695e-f8da-4abb-9a6e-a24d66c6b633%2Fh83ylz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Understanding the Lapse Rate in Meteorology: An Application of the Mean Value Theorem**
The lapse rate is the rate at which the temperature decreases in the atmosphere with respect to increasing altitude. Concurrent measurements indicate that at an elevation of 6.6 km, the temperature is -8.4°C, and at an elevation of 3.5 km, the temperature is 9.6°C. Based on the Mean Value Theorem, can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation?
---
### Analysis Question:
**Can it be concluded that the lapse rate exceeds the threshold value of 3°C/km at some intermediate elevation?**
---
**Answer Choices:**
- **A. Yes.** At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value.
- **B. No.** At some point between 3.5 km and 6.6 km, the lapse rate is 5.8°C/km, which is greater than the threshold value.
- **C. Yes.** At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
- **D. No.** At some point between 3.5 km and 6.6 km, the lapse rate is 0.2°C/km, which is less than the threshold value.
---
### Explanation:
To determine whether the lapse rate exceeds the threshold value of 3°C/km at any point between the given altitudes, we apply the Mean Value Theorem. This theorem states that if a function is continuous on the closed interval \([a, b]\) and differentiable on the open interval \((a, b)\), then there exists at least one \(c\) in the interval \((a, b)\) such that the derivative at \(c\) equals the average rate of change of the function over \([a, b]\).
Let's calculate the average rate of change (lapse rate) between the altitudes 3.5 km and 6.6 km:
1. **Temperature at 6.6 km:** -8.4°C
2. **Temperature at 3.5 km:** 9.6°C
3. **Change in temperature
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