Q5. The volume of an embankment having a total length of L and cross-section areas A₁, A₂, A₃, A₄ at an interval of H using the trapezoidal method is:
📚 Detailed Explanation: Trapezoidal Volume Formula for 4 Cross-Sections
The trapezoidal rule for volumes treats the solid between consecutive cross-sections as a prism whose volume is (average of the two end areas) × (distance between them). When multiple sections exist at equal spacing H, the formula generalises elegantly: add half of the first and last areas plus all intermediate areas, then multiply by H.
V = H × [(A₁ + Aₙ)/2 + A₂ + A₃ + … + Aₙ⁻₁]
For n = 4 sections (A₁, A₂, A₃, A₄), the first section is A₁ and the last is A₄:
V = H × [(A₁ + A₄)/2 + A₂ + A₃]
Derivation from First Principles
Between sections 2 and 3: V₂₃ = H × (A₂ + A₃)/2
Between sections 3 and 4: V₃₄ = H × (A₃ + A₄)/2
Total V = Vâ‚â‚‚ + V₂₃ + V₃₄
= H × [(Aâ‚+Aâ‚‚)/2 + (Aâ‚‚+A₃)/2 + (A₃+Aâ‚„)/2]
= H × [Aâ‚/2 + Aâ‚‚ + A₃ + Aâ‚„/2] ↠Aâ‚‚ and A₃ each appear twice, halved twice = once
= H × [(Aâ‚+Aâ‚„)/2 + Aâ‚‚ + A₃] ✓ matches option A
Why the Other Options Are Wrong
B: Uses (A₁+A₄)/4 instead of /2. Halving the end-area contribution again underestimates the volume — there is no mathematical basis for dividing by 4 in the trapezoidal rule.
C: Uses L (total embankment length) instead of H (interval between sections). If there are 4 sections at equal spacing H over total length L = 3H, then L ≠ H. Using L gives a result 3 times too large. The multiplier must be the interval H, not the total length L.
D: This is a garbled mix of the prismoidal formula structure and the trapezoidal formula — the expression inside is not a valid volume formula for any standard method.
Key Concepts for Students
- H vs L: H is the uniform interval between successive cross-sections, not the total length. For 4 sections equally spaced, total length L = 3H (three gaps). Always use H as the multiplier, never L.
- The pattern to remember: In the trapezoidal rule, end areas get coefficient ½, all middle areas get coefficient 1. In Simpson’s prismoidal rule, end areas get coefficient 1, odd-position middle areas get 4, even-position middle areas get 2, and the whole sum is multiplied by d/3.
- Practical use: For a road cutting or embankment, cross-sections are surveyed at regular intervals (say every 20 m). You compute each section’s area, then apply the trapezoidal formula to get total volume, and from that calculate the cost of excavation or fill.