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

$500L,$ and increases at a constant feed rate $F=40\frac{L}{h}.$ Under these conditions, the biomass concentra$-$

tion is constant. Also, the biomass growth follows Monod kinetics with ${}_{\text{m}\text{a}\text{x}}=0\mathrm{.}5h^{-1}$ and $K_s=12\frac{g}{L}.$

$($a$)$ Derive a relation between the specific growth rate $$ and the volume $V$ and change in volume $F.$

$($b$)$ Calculate the value of $$ after $10$ hours.

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