For a circular channel flowing partially full, select the correct set of design facts: depth for maximum velocity, hydraulic mean depth for maximum velocity, and depth for maximum discharge.

Introduction / Context:
Hydraulic proportioning of circular channels (sewers, tunnels) uses known optima for velocity and discharge. These conditions depend on the filling depth relative to diameter and on how area and wetted perimeter combine to produce hydraulic radius and velocity under a given slope and roughness.

Given Data / Assumptions:

• Circular section of diameter D, flowing partially full under gravity.
• Chezy/Manning relations apply; roughness and slope are fixed during comparisons.
• Comparison across relative depths only.

Concept / Approach:
Velocity depends on hydraulic radius R and hydraulic radius increases with certain filling depths. Discharge depends on both area and velocity. Standard derivations using calculus on the geometric relations for A(θ) and P(θ) yield approximate optima: y/D ≈ 0.81 for maximum velocity, R ≈ 0.30 D at that state, and y/D ≈ 0.95 for maximum discharge.

Step-by-Step Solution:
Express A and P in terms of central angle and y/D.Compute R = A/P and use V ∝ R^(2/3) S^(1/2) (Manning) or V ∝ C √(R S).Differentiate with respect to y/D to locate maxima for V and Q = A V.

Verification / Alternative check:
Published design charts and handbooks confirm the canonical values y/D ≈ 0.81 for maximum V and y/D ≈ 0.95 for maximum Q for circular sections.

Why Other Options Are Wrong:
Other numerical sets do not match the known extrema and would lead to under- or over-sized channels.

Common Pitfalls:

• Assuming the same depth maximizes both velocity and discharge—these are different optima.
• Using full-pipe formulas for partially full conditions.

Final Answer:
Depth for maximum velocity ≈ 0.81 D; hydraulic mean depth ≈ 0.30 D; depth for maximum discharge ≈ 0.95 D