$(20$ pts$)$ A microorganism is grown in a chemostat using sterile feed containing $8\frac{g}{L}$ of the limiting carbon substrate. The following exit substrate concentrations are observed at different dilution rates:

$\backslash$table$[[D(h^{-1}),0\mathrm{.}05,0\mathrm{.}20,0\mathrm{.}40,0\mathrm{.}60,0\mathrm{.}80],[S(m\frac{g}{m}L),0\mathrm{.}01,0\mathrm{.}05,0\mathrm{.}13,0\mathrm{.}35,2\mathrm{.}25]]$

Assuming simple Monod growth, determine $($a$)$ the maximum specific growth rate $({}_m),$ and the saturation constant $(K_s)$; and $($b$)$ show that the assumption of simple Monod kinetics used in part $($a$)$ is valid.