Question

The infiltration capacity of a basin is described by Horton's equation: \[ I = 2 + e^{-3t} \] where \(I\) is in cm/h and \(t\) is in hours. If the duration of the storm event is 2 hours, the depth of infiltration in the last 1 hour of the storm event in mm is:

• A. 5
• B. 10
• C. 20
• D. 25

Hint:

For Horton's equation, integrate over the desired duration. Don't forget to convert cm → mm for final answers.

Answer 1

The Correct Option is C

Step 1: Horton's infiltration equation. \[ I(t) = 2 + e^{-3t} \text{cm/h} \]

Step 2: Total infiltration between \(t_1\) and \(t_2\). \[ F = \int_{t_1}^{t_2} I(t) \, dt \] For last 1 hour: \(t_1 = 1, \; t_2 = 2\). \[ F = \int_1^2 (2 + e^{-3t}) dt \] \[ = \int_1^2 2 dt + \int_1^2 e^{-3t} dt \] \[ = 2(t)\big|_1^2 + \frac{e^{-3t}}{-3}\Big|_1^2 \]

Step 3: Evaluate integrals. \[ = 2(2-1) + \left( \frac{e^{-6} - e^{-3}}{-3} \right) \] \[ = 2 + \frac{(0.00248 - 0.0498)}{-3} \] \[ = 2 + \frac{-0.0473}{-3} = 2 + 0.0158 = 2.016 \, cm \]

Step 4: Convert to mm. \[ 2.016 \, cm = 20.16 \, mm \approx 20 \, mm \] \[ \boxed{20 \, mm} \]