A contractor is estimating the volume of soil to be excavated for a $500 \text{ m}$ long highway cut. The cross-sectional areas at each $20 \text{ m}$ station vary significantly because the terrain goes from a flat plain into a sharp, irregular hillside.

The contractor's junior engineer calculated the total volume using the Average End-Area Method, but the senior engineer rejected the calculation and demanded they recalculate using the Prismoidal Formula.

Explain why the senior engineer rejected the Average End-Area Method for this specific terrain.

A highway contractor is excavating 10,000 cubic meters (bank volume) of dense, undisturbed clay from a cut section. They plan to use this exact volume to fill a 10,000 cubic meter depression (compacted volume) further down the road. The clay has a swell factor of 25% upon excavation and a shrinkage factor of 10% when heavily compacted compared to its original bank state.

Explain why this plan will fail, and calculate the actual volume of compacted fill the cut will produce.

A highway route has three consecutive cross-sections spaced $30 \text{ m}$ apart ($L = 30 \text{ m}$). The areas of the cross-sections are:

• Station $10+000$: $A_1 = 45.5 \text{ m}^2$
• Station $10+030$: $A_2 = 60.2 \text{ m}^2$
• Station $10+060$: $A_3 = 85.0 \text{ m}^2$

Calculate the total volume of earthwork between Station 10+000 and Station 10+060 using the Average End-Area method.

Using the exact same data from Problem 1 ($A_1 = 45.5 \text{ m}^2$ at $0 \text{ m}$, $A_2 = 60.2 \text{ m}^2$ at $30 \text{ m}$, and $A_3 = 85.0 \text{ m}^2$ at $60 \text{ m}$), calculate the total volume across the entire $60 \text{ m}$ stretch as a single prismoid using the Prismoidal Formula.

The End-Area volume of a $20 \text{ m}$ long earthwork segment ($L=20 \text{ m}$) was calculated as $1200 \text{ m}^3$. The end sections are level sections (flat ground). At Station 1, the center height ($c_1$) is $2.0 \text{ m}$ and the top width ($w_1$) is $14.0 \text{ m}$. At Station 2, the center height ($c_2$) is $4.0 \text{ m}$ and the top width ($w_2$) is $22.0 \text{ m}$.

Calculate the Prismoidal Correction ($C_p$) and use it to find the true (Prismoidal) volume.

A borrow pit is laid out as a $2 \times 2$ grid of squares, where each square measures $15 \text{ m}$ by $15 \text{ m}$. The grid has 3 rows and 3 columns of stakes (total 9 stakes). The depth of excavation (cut) at each stake in meters is given as follows:

• Row 1: $h_{1,1}=1.2$, $h_{1,2}=1.5$, $h_{1,3}=1.8$
• Row 2: $h_{2,1}=1.4$, $h_{2,2}=2.0$, $h_{2,3}=1.9$
• Row 3: $h_{3,1}=1.1$, $h_{3,2}=1.6$, $h_{3,3}=1.3$

Calculate the total volume of earth excavated from the borrow pit.