Most economical trapezoidal channel section — identifying equivalent criteria: Which of the following conditions are satisfied by a hydraulically most economical trapezoidal open-channel section?

Introduction / Context:
The “most economical” (best hydraulic) section minimizes wetted perimeter for a given area, thereby maximizing discharge for a given slope and roughness (or minimizing energy loss). For trapezoidal channels, several geometric conditions are equivalent indicators of this optimum shape.

Given Data / Assumptions:

• Uniform, steady, open-channel flow.
• Trapezoidal cross-section with base width b, side slope m:1, and depth y.
• Optimization under constant area and slope (Manning/Chezy framework).

Concept / Approach:

For the most economical trapezoid, the condition is that the hydraulic radius R = A/P is maximized. This leads to canonical geometric results that can be expressed in multiple equivalent forms, including relationships among top width, side lengths, and inscribed semicircle tangency to the wetted boundary.

Step-by-Step Solution:

Verification / Alternative check:

Textbook derivations using Chezy or Manning demonstrate identical optimum conditions; numerical examples confirm lower P for the same A at the optimum.

Why Other Options Are Wrong:

Here, (a), (b), and (c) are each true descriptors; thus selecting only one would be incomplete. The correct comprehensive choice is (d).

Common Pitfalls:

Confusing the most economical rectangle (b = 2y) with trapezoid conditions; overlooking that several geometric tests are equivalent at optimum.

Final Answer:

All of these