Question

A canal supplies water to an area growing wheat over 100 hectares. The duration between the first and last watering is 120 days, and the total depth of water required by the crop is 35 cm. The most intense watering is required over a period of 30 days and requires a total depth of water equal to 12 cm. Assuming precipitation to be negligible and neglecting all losses, the minimum discharge (in m\(^3\)/s, rounded off to three decimal places) in the canal to satisfy the crop requirement is \underline{\hspace{2cm}.}

Hint:

To calculate the discharge, remember that volume = area \(\times\) depth. For discharge, divide the volume by the time period (in seconds) for which the water is being supplied.

Answer 1

Step 1: Convert hectares to square meters.
The area is given as 100 hectares. Since 1 hectare = 10,000 m\(^2\), the area is: \[ 100 \, \text{hectares} = 100 \times 10,000 \, \text{m}^2 = 1,000,000 \, \text{m}^2 \]

Step 2: Convert depth to meters.
The total depth of water required by the crop is 35 cm, which equals 0.35 m. The depth of water required during the most intense watering is 12 cm, which equals 0.12 m.

Step 3: Calculate total volume of water required.
The total volume of water required for the crop over the entire area (100 hectares) is the product of the area and the depth of water: \[ \text{Total volume required} = \text{Area} \times \text{Depth} = 1,000,000 \, \text{m}^2 \times 0.35 \, \text{m} = 350,000 \, \text{m}^3 \]

Step 4: Calculate the volume required for the most intense watering period.
The volume of water required during the most intense 30-day period (with a depth of 0.12 m) is: \[ \text{Volume for intense watering} = \text{Area} \times \text{Depth for intense watering} = 1,000,000 \, \text{m}^2 \times 0.12 \, \text{m} = 120,000 \, \text{m}^3 \]

Step 5: Calculate the minimum discharge required.
The minimum discharge is the volume of water needed for the most intense period (30 days) divided by the time (in seconds) over which it needs to be delivered. The total time for the most intense watering period is 30 days, or \( 30 \times 24 \times 60 \times 60 = 2,592,000 \) seconds. Thus, the minimum discharge is: \[ \text{Minimum discharge} = \frac{\text{Volume for intense watering}}{\text{Time}} = \frac{120,000 \, \text{m}^3}{2,592,000 \, \text{s}} = 0.0463 \, \text{m}^3/\text{s} \] \[ \boxed{\text{The minimum discharge required is 0.046 m}^3/\text{s}.} \]