Question

When \(\Delta\) (delta) is in cm, B (base period) is in days and D is in ha cumec\(^{-1}\), the correct relation is:

• A. \(\Delta = \dfrac{864\,B}{D}\) (cm)
• B. \(\Delta = \dfrac{864\,D}{B}\) (cm)
• C. \(\Delta = \dfrac{8640\,B}{D}\) (cm)
• D. \(\Delta = \dfrac{86.4\,B}{D}\) (cm)

Hint:

Start from the fact that 1 cumec running for 1 day covers 1 hectare to a depth of 8.64 metres, then convert to centimetres.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need the correct formula linking delta (\(\Delta\), the total depth of irrigation water used by a crop), the base period B (the crop's growing duration in days), and the duty D (the area in hectares that one cumec of continuous flow can irrigate over that base period), with \(\Delta\) expressed in centimetres.

Step 2: Key Formula or Approach:
Start from the definition of one cumec-day of water. A flow of 1 cumec (1 cubic metre per second) running for one full day supplies \(1 \times 86400 = 86400\) cubic metres of water, since there are 86400 seconds in a day.
One hectare equals 10000 square metres, so spreading 1 cumec-day of water evenly over 1 hectare gives a depth of \(86400 / 10000 = 8.64\) metres. This 8.64 is the well known constant linking cumec-days to hectare-metres.

Step 3: Detailed Explanation:
For a duty of D hectares per cumec applied continuously over a base period of B days, the total water supplied is \(B\) cumec-days, and this water irrigates D hectares.
The depth applied, in metres, is total cumec-days times 8.64, divided by the area in hectares:
\[ \Delta(\text{m}) = \frac{8.64\,B}{D} \]
Since the question asks for \(\Delta\) in centimetres, and 1 metre equals 100 centimetres, multiply both sides by 100:
\[ \Delta(\text{cm}) = \frac{8.64 \times 100 \, B}{D} = \frac{864\,B}{D} \]
This matches option 1 exactly. Option 4 keeps the constant as 86.4, which is actually the metre form multiplied by only 10, not 100, so it is out by a factor of ten. Option 3 uses 8640, ten times too large, and option 2 wrongly flips B and D in the fraction.

Step 4: Final Answer:
\[ \boxed{\Delta = \dfrac{864\,B}{D} \text{ (cm)}} \]