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A wastewater treatment system using activated sludge needs to lower $BO{D}_5$ levels from $1000m\frac{g}{l}$ to $40m\frac{g}{l}$ at the discharge. The biomass is concentrated threefold in the sedimentation unit. The kinetic parameters are ${}_m=0\mathrm{.}4h^{-1},K_s=80m\frac{g}{l},k_d=0\mathrm{.}02h^{-1},$ and $Y_{\frac{x}{S}}^M=0\mathrm{.}5gMLVS\frac{S}{g}BO{D}_5.$ The wastewater flow rate is $30000\frac{\text{I}}{h},$ and the treatment basin has a volume of $50,000$I.

$(5$ Marks$)$

$($a$.)$ What is the value of the solid's residence time $($i$.$e$.,{}_c)?$

$($b$.)$ What value of the recycle ratio must be used?