For the most economical (best hydraulic) trapezoidal open-channel section, which geometric condition must hold true? Choose the classical proportionality used in design.

Introduction / Context:The “most economical” or “best hydraulic” section minimizes wetted perimeter for a given area, thereby maximizing hydraulic radius and reducing friction losses. For trapezoidal channels used in irrigation and drainage, a celebrated condition ties the top width and the sloping side length.

Given Data / Assumptions:

• Prismatic trapezoidal channel with side slope z:1 (horizontal:vertical).
• Goal: minimize wetted perimeter P for a given flow area A.
• Steady, uniform open-channel flow concept (e.g., Manning/Chezy context).

Concept / Approach:

Using calculus of variations or standard derivations, the conditions for the best trapezoidal section are: (1) hydraulic radius R = A/P = y/2, and (2) half of the top width equals the sloping side length. This condition directly shapes the cross section and simplifies design checks.

Step-by-Step Solution:

Verification / Alternative check:

Applying the condition returns the known result R = y/2 at optimum, consistent with minimal P for a given A.

Why Other Options Are Wrong:

(a) and (c) are not general optimality conditions. (e) is not a standard best-section identity. Only (b) matches the classical economical trapezoid result.

Common Pitfalls:

Confusing the optimal trapezoid with the rectangular or semicircular best sections; each has different characteristic ratios.

Final Answer:

Half of the top width equals one sloping side length