The table below summarizes rainfall data for a storm event in a $30-$acre watershed in Midland, TX$.$ The total runoff during the storm was $130,000f{t}^3.$ The watershed soil is capable of infiltrating at a rate of $3\mathrm{.}0i\frac{n}{h}r$ at the beginning of the storm. By the end of the storm, it can only infiltrate at $\frac{1}{3}$ of the initial rate. Assume the decay constant is $0\mathrm{.}4h{r}^{-1}.$

$\backslash$table$[[$Time $($hr$),$Rainfall $($in$/$hr$)],[1,0\mathrm{.}8],[2,1\mathrm{.}0],[3,2\mathrm{.}4],[4,2\mathrm{.}8],[5,3\mathrm{.}1],[6,2\mathrm{.}0],[7,1\mathrm{.}8],[8,1\mathrm{.}2]]$

$($a$)$ Use the $$ index method to find the total infiltration during the storm event $($in inches$).$

$($b$)$ Use Horton's method to find the time $($after the start of the storm$)$ when the infiltration rate is approximately half of the initial rate.

$($c$)$ Use Horton's method to find the total infiltration during the storm event $($in inches$).$

$($d$)$ How do the methods from parts $($a$)$ and $($c$)$ differ? Which do you think is a more accurate estimate of actual infiltration during the storm? Why?