$(15$ points$)$ A $50{10}^7\frac{L}{?}$ day activated sludge plant is being designed to treat domestic

wastewater. The operators aim to remove $90\%$ of the $BO{D}_5$ entering the aeration basin. The

following operational design parameters are chosen: the mixed liquor volatile suspended solids

$($MLVSS$)$ concentration in the aeration basin is $2,200m\frac{g}{L}$; MLVSS concentration in the waste

sludge is $3500m\frac{g}{L}$; influent $BO{D}_5$ concentration is $200m\frac{g}{L}$ of which $35\%$ is removed by

primary clarification. The design SRT is $5$ days, and sludge is processed on the belt filter very

week. The microorganisms in the activated sludge process can convert $100g$ of $BO{D}_5$ into $55g$

biomass. They have a maximum growth rate of $\frac{0\mathrm{.}1}{?}$ day and a first$-$order death rate constant of

$\frac{0\mathrm{.}05}{?}$ day, and they reach half of their maximum growth rate when the $BO{D}_5$ concentration is $10$

$m\frac{g}{L}.$ What is the total aeration tank volume required $(m^3)?$