Problem: Calculate the cross-sectional area of a fill section using the coordinate method. The coordinates $(x, y)$ of the vertices relative to the centerline (where $x$ is horizontal offset and $y$ is vertical elevation from a datum) are given as follows: $(0, 5), (10, 5), (15, 0), (-15, 0), (-10, 5)$.

Problem: A $100\text{ m}$ segment of a highway requires excavation (cut). The cross-sectional area at station $10+00$ ($A_1$) is $45\text{ m}^2$. The cross-sectional area at station $11+00$ ($A_2$) is $65\text{ m}^2$. Calculate the volume of earthwork using the Average End Area method.

Problem: Using the same stations from Example 2, suppose a detailed mid-section survey at station $10+50$ reveals the true mid-area $A_m$ is $50\text{ m}^2$. Calculate the volume of earthwork using the more accurate Prismoidal formula.

Problem: A road segment is $100\text{ m}$ long. Station 1 has a top width $w_1 = 20\text{ m}$ and a centerline height $h_1 = 4\text{ m}$. Station 2 has a top width $w_2 = 30\text{ m}$ and a centerline height $h_2 = 6\text{ m}$. Calculate the Prismoidal Correction ($C_p$).

Scenario: A contractor is analyzing a $5\text{ km}$ highway project. The initial section ($0$ to $1.5\text{ km}$) requires significant cut through a hill. The middle section ($1.5$ to $3\text{ km}$) crosses a valley requiring fill. The final section ($3$ to $5\text{ km}$) is mostly flat, requiring minor fill.

Scenario: The contract specifies a freehaul distance of $500\text{ m}$. For any material moved beyond this distance, the contractor is paid an overhaul rate of $$2.50$per cubic meter-station (where a station is$100\text$).