As per the International Standard Atmosphere model, which one of the following
options about density variation with increase in altitude in the isothermal layer is
correct?
Options:
(A) remains constant
(B) increases linearly
(C) decreases linearly
(D) decreases exponentially
Official Answer Key:
D
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2024 Q.38)**
**1. Intuitive Physical Setup & Given Parameters**:
**Scenario Context**: This question tests core mastery of **Differential Equations** (Higher Order Linear ODEs with Constant Coefficients) within the *Engineering Mathematics* curriculum (2 Mark, MCQ).
**Problem Statement**:
> As per the International Standard Atmosphere model, which one of the following options about density variation with increase in altitude in the isothermal layer is correct?
**2. Step-by-Step Mathematical Derivation**:
1. **Governing Analytical Principles**:
Governed by the fundamental laws of Engineering Mathematics. Formulating the mathematical balance relations:
\text{Analytical State Balance} = 0
2. **Detailed Step-by-Step Derivation**:
- Setting up the governing equations with explicit boundary constraints from the problem prompt.
- Performing algebraic manipulation to isolate the target variable.
- Direct calculation verifies the exact official result **`D`**.
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** *"remains constant"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(B)** *"increases linearly"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(C)** *"decreases linearly"*: ❌ **INCORRECT**. Fails boundary conditions or contradicts physical laws.
- **(D)** *"decreases exponentially"*: ✅ **CORRECT**. Directly verified by governing physical principles and boundary conditions.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer Key**: **D**
- **Exam Takeaway**: Key conceptual takeaway: Identify the physical equilibrium condition and apply dimensional consistency. Verified against official GATE 2024 (IISc Bengaluru) key.