$.(30$ points$)$ A printed circuit board manufacturing plant is discharging the chemical$1,1,1-$trichloroethane $($TCA$)$ into a river that subsequently flows into a lake, the source of drinking water for a small town. The levels of TCA in this lake currently fail to meet drinking water objectives of $10$ppb$.$ As an environmental specialist with the state department of environmental protection, you have been asked to help determine discharge requirements for the manufacturing plant. By assuming that the lake is a well$-$mixed system, that concentrations are at steady state, and that the only processes acting on this chemical in the lake are volatilization and biodegradation, how high can the influent concentration of TCA $($Cin$)$ be at the point where the river enters the lake? Some pertinent data are: TCA volatilization rate: $3*10^-4$ microgram$/$cm$^2$ sec;TCA biodegradation rate: $6*10^-3$ micromole$/$L$.$day; River inflow to lake: $200$ gal$/$min;River outflow from lake:$200$ gal$/$min;Lake volume:$1\mathrm{.}5*10^6$gal;Average lake depth:$10$ft