GATE 2024 CE (CE2) – Question 56
A storm with a recorded precipitation of 11.0 cm, as shown in the table, produced a direct run-off of 6.0 cm.
| Time from start (hours) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
|---|---|---|---|---|---|---|---|---|
| Recorded cumulative precipitation (cm) | 0.5 | 1.5 | 3.1 | 5.5 | 7.3 | 8.9 | 10.2 | 11.0 |
The $\phi$-index of this storm is ____________ cm/hr (rounded off to 2 decimal places).
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Correct answer: 0.63 to 0.65
Explanation
The hourly rainfall is 0.5, 1.0, 1.6, 2.4, 1.8, 1.6, 1.3 and 0.8 cm. The total loss is $11.0 - 6.0 = 5.0$ cm. Assume that the rainfall exceeds the $\phi$-index in hours 2 to 8, which is 7 hours, and that hour 1 (0.5 cm) is all loss. Then the loss is $0.5 + 7\phi$... more simply the runoff is $\sum(p_i - \phi)$ over the hours with $p_i > \phi$: $(11.0 - 0.5) - 7\phi = 6.0$, so $\phi = \frac{4.5}{7} = 0.643$ cm/h. This is above 0.5 (hour 1 is below $\phi$) and below 0.8 (all other hours are above), so the assumption is consistent.