a$.$ What is the ultimate BOD $(\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ))$ of wastewater discharge?

b$.$ Assuming complete and instantaneous mixing, find the ultimate BOD $(\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ))$ of the mixture of waste and river just downstream from the outfall $($or just after the mixing point$).$

c$.$ Assuming a constant cross$-$sectional area for the stream equal to $\backslash (55\backslash$mathrm$\{$~m$\}^\{2\}\backslash ),$ what ultimate BOD $(\backslash (\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash ))$ would you expect to find at a point $10$ km downstream?

d$.$ EPA found that BOD discharged at the mixing point exceeds the permitted value and asked the wastewater treatment plant owner to meet the concentration limit of $\backslash (20\backslash$mathrm$\{$mg$\}/\backslash$mathrm$\{$L$\}\backslash )$ at the mixing point after discharge. It is expensive to modify the wastewater plant, so you are assigned to design a detention pond that receives the wastewater effluent and reduce the ultimate BOD discharged to the stream. What would be the ultimate BOD of wastewater discharged from the pond to stream so that the ultimate BOD after mixing would meet the EPA permitted value?

e$.(2$ pt$)$ What should be the size $\backslash (\backslash$left$(\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$)\backslash )$ of the detention pond, so that the ultimate BOD after mixing would meet the EPA permitted value? $($assume the detention pond is a CMFR and BOD decays at the same rate, $\backslash (\backslash$mathrm$\{$k$\}=0\mathrm{.}2/\backslash )$ day, in the pond$).$