Q9. In the mid-section formula:
📚 Detailed Explanation: The Mid-Section Formula — All Three Statements Are True
The mid-section method is one of the standard approaches for computing earthwork volume along a linear alignment such as a road or canal. It works by finding a representative cross-section at the midpoint between two surveyed sections and using its area to compute the volume of the intervening solid.
1. Mean depth: Dₑ = (D₁ + D₂) / 2
2. Mid-section area: Aₑ = (B × Dₑ) + (S × Dₑ²), where B = base width, S = side slope ratio
3. Volume: V = Aₑ × L, where L = distance between the two sections
Statement A — Mean Depth Is the Average of Two Consecutive Depths
TRUE. At each of the two surveyed sections, the depth of cutting or filling is known (D₁ and D₂). The mean depth Dₑ = (D₁ + D₂)/2 represents the depth at the imaginary mid-section. This is a simple arithmetic average.
Statement B — Mid-Section Area Uses Mean Depth
TRUE. The area of the mid-section Aₑ is computed using Dₑ as the depth in the standard cross-section area formula. For a trapezoidal cross-section (road cutting or embankment with base B and side slope 1:S):
where:
B = formation width (base of the cutting or embankment)
Dₘ = mean depth = (D₠+ D₂)/2
S = side slope (horizontal distance per unit vertical)
Statement C — Volume = Mid-Section Area × Distance
TRUE. Once the mid-section area Aₑ is found, the volume of earthwork between the two surveyed sections separated by distance L is simply V = Aₑ × L. The mid-section area is treated as representative of the entire length L.
Why “All of the Above” Is Correct
Each of the three statements independently and correctly describes one of the three sequential steps of the mid-section method. All three are established surveying principles. Any answer selecting only A, B, or C individually is incomplete.
Key Concepts for Students
- Mid-section vs trapezoidal: The mid-section method uses the area at mean depth (a single representative section), while the trapezoidal method averages the two end areas directly. The mid-section method is slightly less accurate than trapezoidal for sections with large depth variation but is simpler for field use.
- Accuracy note: The mid-section formula underestimates volume compared to the trapezoidal rule when the section area increases (because the mean depth area is smaller than the true average area for nonlinear cross-sections). The prismoidal formula remains the most accurate.
- All of the above strategy: In surveying MCQs, when all individual statements correctly describe a well-defined formula, “all of the above” is the intended answer. Verify each statement is accurate rather than assuming one is a trick.