An adsorption bed $(40$cm in diameter$)$ is packed with activated carbon as the adsorbent. A feed stream containing $0\mathrm{.}15$wt$\%$ of impurity A enters the column at $1000$k$($g$)/($h$)$r $.$ The adsorption of A on activated carbon follows the Langmuir isotherm: q$\_($A$)=(30\mathrm{.}15$c$\_($A$))/(1+5$c$\_($A$))$ where c$\_($A$)$ is the concentration of A in the wastewater stream $($ kg of $($A$)/($k$)$g of wastewater feed$),$ and q$\_($A$)$ is the corresponding maximum amount of impurity A that can be adsorbed $($ kg of $($A$)/($k$)$g adsorbent$).$ Impurity A is first detected in the exiting stream after $15$ hours of operation, and the bed reaches complete saturation after $2$ more hours. $($a$)$ Starting from Langmuir isotherm expression, determine the amount $($in kg $)$ of adsorbent material needed for the bed to achieve saturation after $15$ hours of operation. $($b$)$ Given that the density of activated carbon is $550$k$($g$)/($m$^(3)),$ what is the length $($in m $)$ of the adsorbent bed? $($c$)$ Suppose the breakthrough curve is a standard, S$-$shaped curve, what is the percentage of bed that is not used at breakthrough?