In earthwork volume computation, the trapezoidal formula relies on which fundamental assumption(s) about the geometry of the end sections and area variation?

Introduction / Context:
In highway, canal, and railway earthwork, volumes between two measured cross-sections are commonly approximated using either the trapezoidal (average end area) formula or the more accurate prismoidal formula. Knowing the precise assumptions behind each method is crucial to select the right formula and to apply corrections properly.

Given Data / Assumptions:

• We are comparing statements about end-section geometry and how areas vary between stations.
• Focus is on the basis of the trapezoidal formula, not the prismoidal formula.
• Understanding of prismoidal correction is relevant but must be correctly stated.

Concept / Approach:
The trapezoidal (average end area) formula assumes linear variation of area between two parallel end sections, leading to volume V_trap = (L / 2) * (A1 + A2). Its core geometric prerequisite is that the two end sections are comparable and effectively “parallel planes.” The prismoidal formula V_pris = (L / 6) * (A1 + 4 * A_m + A2) is exact for prismoids where area varies quadratically with distance. Typically, the trapezoidal formula underestimates volume when the mid-area exceeds the mean of end areas; a “prismoidal correction” is added to trapezoidal results to approach prismoidal accuracy, not subtracted from an “over-estimated prismoidal volume”.

Step-by-Step Solution:
Is parallelism of end sections required? Yes — essential for consistent area interpolation in trapezoidal method.Is the “mid-area of a pyramid equals half the average of end areas” a general basis? No — it is a special geometric observation and not the fundamental assumption of trapezoidal volume.Is the statement about “prismoidal being over-estimated” correct? No — prismoidal is the more accurate baseline; trapezoidal is the approximation that may require a positive correction.Therefore, only the first assumption is the correct basis for the trapezoidal formula.

Verification / Alternative check:
Compute with an example where A_m > (A1 + A2)/2: V_pris > V_trap, so the correction is positive and added to the trapezoidal result, confirming the logic.

Why Other Options Are Wrong:

• Option B: A niche geometric note; not the general assumption of the trapezoidal method.
• Option C: Factually incorrect regarding over-estimation by the prismoidal formula.
• Option D: Bundles incorrect statements alongside a correct one, so it is wrong.

Common Pitfalls:
Interchanging the roles of “trapezoidal” and “prismoidal”; assuming corrections reduce prismoidal volume; forgetting that trapezoidal is an approximation reliant on linear area variation.

Final Answer:
Only the assumption that end sections are parallel planes is essential for the trapezoidal formula.