Q3. What is the volume of earthwork of constructing a tank excavated in level ground to a depth of 4 m? Top of tank: rectangular, 50 m × 40 m. Side slope = 2:1 (horizontal : vertical).
📚 Detailed Explanation: Tank Volume by the Prismoidal Formula — V = 5461 m³
When a tank or borrow pit is excavated with sloping sides, its cross-section varies from top to bottom. The Prismoidal formula (also called Simpson’s rule for volumes) accounts for this variation by using three cross-sections — top, mid-depth, and bottom — to accurately compute the volume. It gives a more accurate result than simply averaging the top and bottom areas.
Where D = total depth, A&sub1; = top cross-sectional area, A&sub2; = bottom area, Aₑ = mid-section area (at depth D/2).
Side slope 2:1 means for every 1 m of depth, the excavation widens by 2 m on each side.
Step 1 — Find the Three Cross-Sectional Areas
TOP AREA (Aâ‚) — at depth 0:
Length = 50 m, Width = 40 m
A₠= 50 × 40 = 2000 m²
BOTTOM AREA (A₂) — at depth D = 4 m:
Horizontal reduction each side = s × D = 2 × 4 = 8 m
Bottom length = 50 – 2(8) = 50 – 16 = 34 m
Bottom width = 40 – 2(8) = 40 – 16 = 24 m
A₂ = 34 × 24 = 816 m²
MID-SECTION AREA (Aₘ) — at depth D/2 = 2 m:
Horizontal reduction each side = s × 2 = 2 × 2 = 4 m
Mid length = 50 – 2(4) = 42 m
Mid width = 40 – 2(4) = 32 m
Aₘ = 42 × 32 = 1344 m²
Step 2 — Apply the Prismoidal Formula
V = (4/6) × [2000 + 4(1344) + 816]
V = (2/3) × [2000 + 5376 + 816]
V = (2/3) × 8192
V = 5461.33 m³ ≈ 5461 m³
Why the Other Options Are Wrong
A (8866 m³): Likely from using only the top area × depth: 2000 × 4 = 8000 (not even close), or from not reducing the dimensions for slope. Ignoring the sloped sides entirely gives too large a volume.
B (6688 m³): Possibly from the average end-area method: V = (D/2) × (A&sub1; + A&sub2;) = (4/2) × (2000 + 816) = 2 × 2816 = 5632. That gives option C, not B. Option B may arise from incorrectly computing the bottom area with the wrong slope factor.
C (5632 m³): This is the result from the simpler Average End-Area method: V = (D/2) × (A&sub1; + A&sub2;) = 2 × 2816 = 5632 m³. The average end-area method overestimates volume compared to the prismoidal formula for a frustum (tapered solid). The prismoidal formula is the correct approach here.
Key Concepts for Students
- Side slope notation: A slope of 2:1 (H:V) means the cut slopes outward 2 m for every 1 m of depth. At depth d, each side has reduced in dimension by 2d. Both length and width reduce by 2 × 2d = 4d total (2d from each side).
- Prismoidal correction: The prismoidal correction = (D/6) × (c&sub1; − c&sub2;) × (d&sub1; − d&sub2;), where c and d are dimensions of the cross-sections. Applying this correction to the average end-area result gives the same answer as the direct prismoidal formula.
- When to use which formula: Prismoidal formula = most accurate for any three-section solid. Average end-area = quick estimate, always slightly over-estimates for tapered solids. Mid-section formula = most accurate when many sections are available.