Question

The Initial rate of Infiltration of a watershed is estimated as 5.5 cm/hr and the final capacity is 0.5 cm/hr, and the time constant is $0.4h{r}^{-1}$. The total depth (in cm) of infiltration over the 6 hr period is:
(Use Horton's equation)

• A. 14.37

Answer 1

The correct option is A 14.37
We know that,
$f=f_c+(f_0-f_c)e^{-kt}$

Given,
$f_c=5.5cm/hr,f_c=0.5cm/hr,k=0.4h{r}^{-1}$

$f=0.5+(5.5-0.5)e^{-0.4t}$

$f=0.5+5e^{-0.4t}$

$\text{Infiltration depth}=\underset{0}{\overset{6}{\int }}fdt$
$=\underset{0}{\overset{6}{\int }}(0.5+5e^{-0.4t})dt$

$={[0.5t+\left(\frac{5}{-0.4}\right)e^{-0.4t}]}_0^6$

$=(0.5\times 6)-\frac{5}{0.4}{e}^{-2.4}+\left(\frac{5}{0.4}\right)e^{-0}$

$=14.37cm$