Lined canal geometry: A triangular-sided channel with a circular bottom of radius D has hydraulic mean depth (R = A/P) approximately equal to which of the following?

Introduction / Context:
Hydraulic mean depth (also called hydraulic radius) R = A / P is a key parameter in uniform flow computations (e.g., Manning or Chezy). For sections involving circular arcs, characteristic values of R are frequently used as approximations when side contributions are modest relative to the curved bottom.

Given Data / Assumptions:

• Bottom is a circular arc with radius D (i.e., diameter 2D if viewing full circle context).
• Triangular side slopes contribute area and wetted perimeter but the section is dominated by the circular bottom arc (typical for many lined canal designs).
• We seek a commonly used approximate value.

Concept / Approach:
For a circular conduit running full and for a semicircular channel, R equals D/4. In many practical lined sections with circular bottoms, the effective hydraulic radius remains close to this characteristic value, especially when side area and perimeter contributions proportionally offset. Thus D/4 serves as a standard approximation used for preliminary checks.

Step-by-Step Solution:
1) Recall benchmark: semicircular section → R = D/4.2) Recognize that adding triangular sides increases both A and P.3) For many standard proportions, the resulting R remains close to D/4, which is accepted for preliminary design.

Verification / Alternative check:
Detailed computation of A and P for the exact geometry confirms values near D/4 for common canal proportions. Designers refine R using exact geometry once side slopes and depths are finalized.

Why Other Options Are Wrong:

• D or D/2: overestimates R for curved-bottom channels.
• D/3, D/5: not standard characteristic values for circular-bottom canal sections.

Common Pitfalls:

• Confusing D as arc radius with pipe diameter; remember R for full circle and semicircle is D/4.
• Using the approximation without later geometry-specific verification.

Final Answer:
D/4.