Irrigation duty–delta–base period relation: If Δ is the depth of water (metres), B is the base period (days), and D is the duty (hectare per cumec), which relation correctly connects these quantities?

Introduction / Context:
The classical irrigation relationship links duty (area irrigated per unit discharge), base period (time), and delta (seasonal depth of water). Correct unit handling is essential in canal capacity design and crop-water calculations.

Given Data / Assumptions:

• Δ in metres of water depth.
• B in days (base period).
• D in hectare per cumec (ha per m^3/s).

Concept / Approach:
One cumec flowing for 1 day delivers 86,400 m^3. Spread over 1 hectare (10,000 m^2), this equals 8.64 m depth. Hence, 1 cumec for B days supplies 8.64 * B m over 1 hectare. Generalizing to D hectare per cumec, the delivered depth is (8.64 * B) / D, which by definition equals Δ.

Step-by-Step Solution:
Volume from 1 cumec in B days = 86,400 * B m^3.Depth over 1 ha = (86,400 * B) / 10,000 = 8.64 * B m.For D ha per cumec, Δ = (8.64 * B) / D ⇒ rearrange to D = 8.64 * B / Δ.

Verification / Alternative check:
Dimensional check: D has units ha per cumec, while the right side has (m * day) / m, but the factor 8.64 embeds unit conversions (86,400/10,000) yielding correct practical units for the standard irrigation identity.

Why Other Options Are Wrong:
Other rearrangements either invert the relation or misplace variables, except Δ = 8.64 * B / D which is equivalent but not the form asked (we asked “which relation correctly connects,” and the canonical answer chosen is D = 8.64 * B / Δ).

Common Pitfalls:

• Confusing hectares with square metres in conversions.
• Dropping the 8.64 factor (86,400/10,000).

Final Answer:
D = 8.64 * B / Δ