$($b$)$ You are assigned as a hydrologist tasked with evaluating the infiltration characteristics

of a specific watershed. The recent rainfall event recorded a total rainfall depth of $50\mathrm{.}8$

$mm,$ occurring over a duration of $1$ hour. Soil data indicates an initial moisture content

of $12\%,$ decreasing to $9\%$ at the time when the soil began to dry out, $15$ minutes after

the onset of rainfall. Utilizing the Horton method parameters, with initial infiltration

rate, $f_0$ of $20\mathrm{.}32m\frac{m}{h}r,$ a decay constant, $k$ of $0\mathrm{.}05$ per hour, and a final infiltration rate

$f_i$ of $5\mathrm{.}08m\frac{m}{h}r,$ compute the infiltration rate at $15$ minute time intervals during the

rainfall event using the Horton equation. Illustrate the results in a plot to show the

decay of infiltration over time.

$[14$ Marks$]$