A fed$-$batch reactor is operated with the following characteristics: At time t $=0$ the culture volume V is

$500$ L$,$ and increases at a constant feed rate F $=40$ L$/$h$.$ Under these conditions, the biomass concentration is constant. Also, the biomass growth follows Monod kinetics with mu max $=0\mathrm{.}5$ h$-1$ and Ks $=12$ g$/$L$.$

$($a$)$ Derive a relation between the specific growth rate

mu and the volume V and change in volume F$.$

$($b$)$ Calculate the value of

mu after $10$ hours.

$($c$)$ What is the substrate concentration in the bioreactor at $10$ hours?