$(15$ points$)$ A wastewater flow with the following characteristics is to be treated in a completely mixed activated$-$sludge system. The reactor is to operate at a biomass concentration of $5000mg$ VSS$/$L and the secondary clarifier is designed to thicken the sludge to $10,000mg$ VSS$/$L$.$ For a mean cell residence time of $8d,$ determine:

a$.$ the required volume of the aeration basin

b$.$ the mass and volume of sludge wasted each day

Flow into reactor $=15,000\frac{m^3}{d}$

Influent BOD$5=150m\frac{g}{L}$

Effluent BOD$5=5m\frac{g}{L}$

$Y=0\mathrm{.}5kg$ biomass $\frac{?}{kg}$ BOD ${?}_5$

$k_d=0\mathrm{.}04d^{-1}$