A contour line joins locations having the same height above the mean sea level. Two contour lines representing heights of 500 m and 550 m are separated by a horizontal distance of 2 km. The average slope (gradient) of the terrain between these two contours is:
Options:
(A) 1 in 20
(B) 1 in 40
(C) 1 in 100
(D) 1 in 200
Official Answer Key:
B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2017 Q.65)**
**1. Intuitive Physical Setup & Given Parameters**:
- Vertical contour interval: $\Delta h = 550\text{ m} - 500\text{ m} = 50\text{ m}$.
- Horizontal distance between contours: $L = 2\text{ km} = 2000\text{ m}$.
**2. Step-by-Step Mathematical Derivation**:
1. **Definition of Terrain Slope (Gradient)**:
\text{Slope} = \frac{\text{Vertical Rise}}{\text{Horizontal Run}} = \frac{\Delta h}{L}
2. **Substituting Given Numbers**:
\text{Slope} = \frac{50\text{ m}}{2000\text{ m}} = \frac{50}{2000} = \frac{1}{40}
Expressed in standard engineering ratio format:
\text{Gradient} = 1 \text{ in } 40
**3. Comprehensive Options & Distractor Analysis**:
- **(A)** 1 in 20: ❌ Incorrect. Double the actual slope.
- **(B)** 1 in 40: ✅ **CORRECT**. $\frac{50}{2000} = \frac{1}{40}$.
- **(C)** 1 in 100: ❌ Incorrect.
- **(D)** 1 in 200: ❌ Incorrect.
**4. Official Master Key & Exam Takeaway**:
- **Official Answer**: **B**
- **Exam Rule**: Gradient = $\frac{\Delta h}{\Delta x} = \frac{50}{2000} = \frac{1}{40}$.