The rainfall observed for five successive days in a catchment is 2, 3, 4, 5 and 6 cm. Compute the direct runoff from the catchment due to this rainfall. The φ-index for the rainfall is assumed to be 2 cm/day.
- A12 cm
- B8 cm
- C16 cm
- D10 cmCorrect
Solution
Concept
With φ-index, the runoff from each day is the rainfall minus φ. If rainfall is less than or equal to φ, the runoff is zero.
Formula
Runoff = Σ (Pi - φ), only for days where Pi > φ
Step-by-step solution
1. φ-index = 2 cm/day. 2. Day 1: 2 - 2 = 0 cm. 3. Day 2: 3 - 2 = 1 cm. 4. Day 3: 4 - 2 = 2 cm. 5. Day 4: 5 - 2 = 3 cm. 6. Day 5: 6 - 2 = 4 cm. 7. Total direct runoff = 0 + 1 + 2 + 3 + 4 = 10 cm.
Explanation
The φ-index is a constant loss rate. Rain above this rate becomes runoff. Adding the excess rain of all five days gives 10 cm.
Why the other options are wrong
A (12 cm): this counts a wrong loss or adds an extra day. B (8 cm): this subtracts too much loss. C (16 cm): this is wrong arithmetic. Total rain is 20 cm and total loss is 10 cm, so runoff is 10 cm, not 16 cm.
Shortcut
Total rain = 20 cm. Loss = 2 × 5 = 10 cm (every day has P ≥ φ). Runoff = 20 - 10 = 10 cm.
Common mistake
Using 2 × 4 = 8 cm as the loss instead of 10 cm, or adding the loss to the rain instead of subtracting it.
Correct answer: D) 10 cm