Practice Question: Volume of Earthwork Due to Excavation

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Solution

First, calculate the area of both trapezoidal cross-sections using the formula for the area of a trapezoid:

\(A = \frac{1}{2} (b_1 + b_2)h\)

where \(b_1\) and \(b_2\) are the lengths of the parallel sides and \(h\) is the distance (height) between the bases.

For the first cross-section \(A_1\):

\(A_1 = \frac{1}{2} (40 \text{ ft} + 20 \text{ ft}) \times 10 \text{ ft}\)
\(A_1 = \frac{1}{2} \times 60 \text{ ft} \times 10 \text{ ft}\)
\(A_1 = 300 \text{ sq ft}\)

For the second cross-section \(A_2\):

\(A_2 = \frac{1}{2} (30 \text{ ft} + 15 \text{ ft}) \times 12 \text{ ft}\)
\(A_2 = \frac{1}{2} \times 45 \text{ ft} \times 12 \text{ ft}\)
\(A_2 = 270 \text{ sq ft}\)

Then, use the average end-area method to calculate the volume:

\(V = L \left( \frac{A_1 + A_2}{2} \right)\)
\(V = 300 \text{ ft} \left( \frac{300 \text{ sq ft} + 270 \text{ sq ft}}{2} \right)\)
\(V = 300 \text{ ft} \times 285 \text{ sq ft}\)
\(V = 85,500 \text{ cubic feet} \approx 3,167 \ yd^3\)

 

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