A culture of bacteria that secrete a product in a chemostat operated at steady state. The specific growth rate of biomass is adequately described by the Monod equation, and the volumetric rate of product formation is described by Leudeking$-$Piret kinetics: dP$/$dt $=(\backslash$alpha $\backslash$mu $+\backslash$beta $)\backslash$Chi

This system is well characterized, such that the following constants are known:

Yx$/$s $=0\mathrm{.}6$ gX$/$gS

$\backslash$alpha $=0\mathrm{.}2$ gP$/$gX

S$0=15$ gS$/$L

$$max$=1\mathrm{.}3$ h$$

$\backslash$beta $=0\mathrm{.}5$ gP$/$gX$-$h

Ks$=0\mathrm{.}3$ g$/$L

Yp$/$s $=1\mathrm{.}4$ gP$/$gS

V$=500$ L

The liquid feed to the chemostat is sterile and the flow rates entering and exiting the chemostat are equal.

$($a$)$ What should the flow rate be adjusted to in this chemostat to achieve the optimal productivity of product formation?

$($b$)$ What is the specific rate of product formation during cell maintenance alone?

$($c$)$ Define the a coefficient in Leudeking$-$Piret kinetics. What is the total specific rate of product formation?