Q7. Areas included by contour lines for a proposed dam: Contour 410 m = 205 ha, 420 m = 120 ha, 430 m = 145 ha, 440 m = 95 ha, 450 m = 135 ha. Calculate the capacity of the dam (in m³) by the trapezoidal method.
📚 Detailed Explanation: Dam Capacity by Trapezoidal Method — V = 53,000,000 m³
The capacity (storage volume) of a dam or reservoir is computed from contour maps. Each contour encloses a horizontal area; the volume between two consecutive contours is found by treating that layer as a trapezoid. The total volume is the sum of all such layers from the deepest contour to the dam crest.
V = d × [(A₁ + Aₙ)/2 + A₂ + A₃ + A₄]
d = contour interval = 420 − 410 = 10 m
A₁ = 205 ha, A₂ = 120 ha, A₃ = 145 ha, A₄ = 95 ha, Aₙ = 135 ha
Step-by-Step Calculation
Areas: Aâ‚=205, Aâ‚‚=120, A₃=145, Aâ‚„=95, Aâ‚…=135 (all in hectares)
V = d × [(A₠+ A₅)/2 + A₂ + A₃ + A₄]
V = 10 × [(205 + 135)/2 + 120 + 145 + 95]
V = 10 × [340/2 + 360]
V = 10 × [170 + 360]
V = 10 × 530
V = 5300 hectare-metres
Unit conversion: 1 hectare = 10,000 m², so 1 hectare-metre = 10,000 m³
V = 5300 × 10,000 = 53,000,000 m³
Why the Unit Conversion Is Critical
| Step | Value | Unit |
|---|---|---|
| Interval d | 10 | m |
| Areas A₁ … Aₙ | 205, 120, 145, 95, 135 | hectares |
| Raw result | 5300 | hectare-metres |
| 1 hectare | = 10,000 | m² |
| 1 hectare-metre | = 10,000 | m³ |
| Final volume | 5300 × 10,000 = 53,000,000 | m³ |
Why the Other Options Are Wrong
A (42,000,000 m³): Could result from adding all five areas directly without halving the end areas: 10 × (205+120+145+95+135) = 10 × 700 = 7000 ha·m = 70,000,000. Doesn’t match; likely from a different wrong combination of areas or wrong contour interval.
C (70,000,000 m³): Exactly what you get if you sum all 5 areas without applying the trapezoidal “half the ends” rule: 10 × 700 ha·m × 10,000 = 70,000,000 m³. This is the classic error of treating all areas with coefficient 1 instead of halving the first and last.
D (80,000,000 m³): Possibly from using d = 10 but adding an extra full area equivalent (e.g., counting one area twice), or from using the wrong contour interval (e.g., 450−410 = 40 m) with a simplified formula.
Key Concepts for Students
- Contour interval as d: The distance between consecutive contours is always measured vertically (elevation difference), not horizontally. Here d = 10 m because successive contours differ by 10 m in elevation.
- Reservoir capacity curve: In practice, dam engineers plot a “capacity curve” (volume vs. elevation) by computing cumulative volumes at each contour level. This allows quick read-off of storage at any water level.
- Simpson’s rule alternative: For 5 contour areas (4 intervals), Simpson’s rule requires an even number of intervals — 4 intervals is even, so Simpson applies: V = (d/3)(A₁ + 4A₂ + 2A₃ + 4A₄ + Aₙ) × 10,000. This gives a slightly different (more accurate) answer than the trapezoidal result.