For a protective embankment with mean height d, side slopes S:1 (horizontal:vertical), and length L, what is the area of one sloping face?

Introduction / Context:
When estimating protection works, pitching, or turfing on embankment slopes, we need the true sloping-face area, not just the vertical projection. Given the mean height, side slope, and length along the embankment, the surface area is found using basic geometry.

Given Data / Assumptions:

• Mean height of embankment face = d.
• Side slope ratio = S:1 (horizontal:vertical).
• Length along the embankment = L.
• We seek the area of a single sloping face.

Concept / Approach:
The sloping length corresponding to vertical height d on a slope S:1 is the hypotenuse of a right triangle with vertical leg d and horizontal leg S*d. Hence, sloping length = d * sqrt(1 + S^2). The surface area of one face equals this sloping length multiplied by the embankment length L.

Step-by-Step Solution:
Compute slope length: l_slope = d * sqrt(1 + S^2).Compute area of one face: A = L * l_slope = L * d * sqrt(1 + S^2).Select the option matching this expression.

Verification / Alternative check:
Dimensional check: L (length) * d (length) gives area (length^2); the sqrt term is dimensionless.

Why Other Options Are Wrong:

• A and B are dimensionally or conceptually incorrect.
• D doubles the correct single-face area (might apply if both faces are considered).
• E is dimensionally inconsistent for area.

Common Pitfalls:
Forgetting to use the true sloping length (using d alone); accidentally doubling for two faces when only one is asked.

Final Answer:
L * d * sqrt(1 + S^2)