This question requires understanding the relationship between key irrigation parameters: Duty, Delta, and Base Period.

Defining Irrigation Terms

• Base Period (B): This is the total duration, measured in days, for which irrigation water is supplied to a crop throughout its growth cycle.
• Duty (D): Duty represents the area of land, measured in hectares, that can be irrigated by a continuously flowing stream of water with a discharge of 1 cubic meter per second (cumec) throughout the base period. Its units are hectares/cumec.
• Delta (Δ): Delta is the total depth of water, measured in meters, required by a crop during its entire base period. It signifies the overall water needs of the crop.

Deriving the Irrigation Relationship Formula

To establish the relationship, we consider the total volume of water required for irrigating a certain area.

Let's assume a discharge Q = 1 cumec.

The area irrigated by this discharge over the base period B (in days) is given by the Duty, D.

Area irrigated, $A = D \times Q$. Since $Q = 1$ cumec, $A = D$ hectares.

We need to express this area in square meters:

$A = D \text{ hectares} \times 10000 \text{ m}^2/\text{hectare} = 10000 \times D \text{ m}^2$.

The total volume of water supplied to this area over the base period B is:

Volume = Area $\times$ Delta

Volume $= (10000 \times D \text{ m}^2) \times (\Delta \text{ m}) = 10000 \times D \times \Delta \text{ m}^3$.

Alternatively, we can calculate the volume based on the discharge and time.

Discharge = 1 cumec = 1 m$^3$/second.

Base period = B days.

Convert the base period to seconds:

Time $= B \text{ days} \times 24 \text{ hours/day} \times 60 \text{ minutes/hour} \times 60 \text{ seconds/minute} = B \times 86400 \text{ seconds}$.

Total Volume supplied = Discharge $\times$ Time

Volume $= (1 \text{ m}^3/\text{second}) \times (B \times 86400 \text{ seconds}) = 86400 \times B \text{ m}^3$.

Now, we equate the two expressions for the total volume:

$10000 \times D \times \Delta = 86400 \times B$

To find the relationship, we rearrange the equation to solve for Delta (Δ):

$\Delta = \frac{86400 \times B}{10000 \times D}$

Simplifying the constant:

$\Delta = \frac{8.64 \times B}{D}$

Conclusion on Relationship

The derived formula clearly shows the direct proportionality between Delta (Δ) and Base Period (B), and the inverse proportionality between Delta (Δ) and Duty (D). This fundamental relationship is crucial in irrigation engineering for estimating crop water requirements and designing irrigation systems.