$2.$ Bed$-$depth service time $($BDST$)$ analysis. A groundwater contaminated with $18$ mg$/$L atrazine has been evaluated for GAC treatment using three pilot columns in series that are $5$ cm in diameter and $3$ m in length at a flow rate of $15$ L$/$min$.$ The pilot data were plotted with line for exhaustion $($C$/$Co$)$ and breakthrough $($C$/$Co$)=0\mathrm{.}10$ having slopes of $4\mathrm{.}87$ days$/$m and $4\mathrm{.}91$ days$/$m and intercepts of $6\mathrm{.}2$ days and $17\mathrm{.}5$ days, respectively. Based on these pilot scale data, determine:

$(1)$ The carbon exhaustion rate $($in Kg$/$day$).$

$(2)$ The time until de carbon was exhausted $($in days$).$

$(3)$ The adsorptive capacity of the carbon $($grams$/$m$3).$

$(4)$ The number of columns required.

$(5)$ The diameter of the full$-$scale pilot system if the flow rate is $7800$ m$3/$day$.$ You can assume that the volumetric loading rate $($m$3/($m$2-$day$)$ of the pilot plant and the full$-$scale plant is the same.

You can assume that the bulk density of GAC is $481$ Kg$/$m$3.$