A valley is $3$ km in width and $13$ km in length. The wind blows at $4\mathrm{.}4$ m$/$s into the valley. The concentration of methane entering the valley is $1$ mg$/$m$3.$ An inversion layer hangs over the town at a height of $120$ m$.$ The degradation of methane in the atmosphere can be described by first order kinetics with a rate constant of $0\mathrm{.}000725$ hr$-1.$ Hydraulic fracturing wells release methane at a rate of $2\mathrm{.}6$ g$/$s and there are $1873$ in the valley. Methane is also released from the three confined animal feeding operations $($CAFOs$)$ in the valley. Each CAFO has an average of $1240$ cattle. Each cattle emits methane at a rate of $15$ g$/$hr$.$

a$)$ On the answer sheet, draw a diagram showing the system and flow of methane into and out of the valley

b$)$ Write the general mass balance equation describing the fate and transport of methane in the valley.

c$)$ State your assumptions in constructing the mass balance equation and determining the methane concentration

d$)$ Determine the steady$-$state concentration in mg$/$m$3.$

e$)$ If the hydraulic fracking wells suddenly ceased operation but the CAFOs still operated, determine the concentration $($in mg$/$m$3)$ of methane after $7$ days.