b$)$ A completely mixed, conventionally aerated activated sludge system is to be used for secondary treatment of a flow of $5,000$ m$3$ d$-1$ of municipal wastewater. After primary sedimentation, the BOD$5$ is $175$ mg L$-1.$ It is required that the BOD$5$ in the final effluent is less than or equal to $10$ mg L$-1.$ Pilot plant studies have established the following kinetic values for the treatment process: Y $=0\mathrm{.}5$ kg kg$-1$ and kd $=0\mathrm{.}05$ d$-1,$ where Y is the decimal fraction of food mass converted to biomass and kd is the decay rate constant. Assuming a mixed liquor suspended solids concentration of $3,320$ mg L$-1$ in the activated sludge reactor, an underflow concentration of $12,000$ mg L$-1$ from the secondary clarifier, and a realistic value for the mean residence time, determine:$($i$)$the volume of the activated sludge reactor. $($ii$)$the mass and volume of solids that must be wasted each day.