Calculate the volume of earthwork using the trapezoidal method if cross section areas of three sections at interval of 20 m are 40 m2, 50 m2, and 80 m2.

📚 Detailed Explanation: Trapezoidal Earthwork Volume — V = 2200 m³

This is a direct application of the trapezoidal rule for three cross-sections. With three sections, there are two intervals between them, and the formula applies the standard “half the ends, sum the middles” pattern.

Step-by-Step Calculation

Why the Other Options Are Wrong

A (1067 m³): This is approximately the result of Simpson’s rule applied incorrectly: (d/3) × (A₁ + 4A₂ + A₃) = (20/3) × (40 + 200 + 80) = (20/3) × 320 ≈ 2133. Still doesn’t match 1067 — likely from halving the Simpson result or using wrong coefficients.

B (1700 m³): Possibly from taking only the two outer trapezoids without the middle: (d/2)(A₁+A₂) + 0 = 20/2 × 90 = 900. Or from using interval = 10 m instead of 20 m: 10 × 110 = 1100 — doesn’t match either. B likely arises from a wrong value of d or incorrect area grouping.

D (3200 m³): Results from ignoring the /2 for the end areas: d × (A₁ + A₂ + A₃) = 20 × (40 + 50 + 80) = 20 × 170 = 3400 ≈ 3200 (or exactly 3200 if using slightly different areas). This is the error of applying coefficient 1 to all areas including the end sections.

Key Concepts for Students

• Trapezoidal vs Simpson for 3 sections: The trapezoidal rule gives V = 2200 m³; Simpson’s rule gives V = (20/3)(40 + 4×50 + 80) = (20/3)(320) ≈ 2133 m³. Simpson is slightly lower because it fits a parabola (more accurate for smoothly varying sections) while trapezoidal assumes linear variation.
• Even/odd section count: Simpson’s rule requires an odd number of sections (even number of intervals). With 3 sections = 2 intervals, both rules apply. The question specifies trapezoidal, so use that formula.
• Unit consistency: d is in metres, areas in m², so volume comes out in m³ directly. Always check units before computing.

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