$25$ points$)$ Given the following information for an E$.$ coli culture producing biomass $(\backslash$mu max $=1\mathrm{.}0$ hr$-1)$: S$0=10$ g$/$L$,$ S $=0\mathrm{.}1$ g$/$L$,$ Xm $=2$ g cells$/$L$,$ X$0=0\mathrm{.}25$ g cells$/$L$,$ kd $=0\mathrm{.}01$ hr$-1$; KS $=0\mathrm{.}01$ g$/$L$,$ tlag $=0\mathrm{.}3$ hr $($D $=0\mathrm{.}2$ hr$-1,$ sterile feed for the chemostat$).$

$($A$)$ Calculate the post$-$batch processing time needed to give a $2$:$1$ ratio of rates of biomass production of continuous versus batch bioreactors $($rc$,$opt$/$rb$).$