The following data was shared by a previous colleague from a batch experiment where an optimised $E.$ coli strain is producing antibiotics. The amount of substrate provided was $1\mathrm{.}75\frac{g}{L}$ reactor.

$\backslash$table$[[$Time $($min$),0,6,8,10,12,13,14,15,17,20],[x($cells$),120,120,120,180,450,690,1100,1800,2500,2600]]$

$($a$)$ Determine the following:

Specific growth rate

Doubling time

Length of the lag phase

$($b$)$ You conduct $3$ follow$-$up experiments with different substrate concentrations and obtain the following data.

$\backslash$table$[[$Time $($min$),0,6,8,10,12,13,14,15,17,20],[S=1,120,120,120,240,420,570,780,1100,2000,2000],[S=5,120,120,120,280,1200,2800,6000,14000,30000,32000],[S=10,120,120,120,310,1500,3400,7500,17000,30000,32000]]$

What is the Michaelis Menten constant for this strain?

$($c$)$ You want to achieve a specific growth rate of $\frac{1}{4}{}_{\text{m}\text{a}\text{x}}$ to optimise the antibiotics production in a $10L$ continuous culture reactor $($chemostat$).$ What should your flow rate into the chemostat and concentration of substrate in the feed be to achieve this?

$($d$)$ You've discovered the following data from a previous experiment in trying to optimise the substrate feed for antibiotics production:

Specific substrate consumption rate $()=0\mathrm{.}6h{r}^{-1}$

Maintenance requirements $(m)=0\mathrm{.}002mi{n}^{-1}$

Yield coefficient of biomass vs substrate $(Y_{\frac{x}{s}})=0\mathrm{.}5$

Experimental data: product vs substrate yield

$\backslash$table$[[$Substrate fed $(\frac{g}{L}),1,1\mathrm{.}75,5,10],[$Product produced $(\frac{g}{L}),0\mathrm{.}2,0\mathrm{.}35,1,2]]$

What would be the specific rate of product formation $q_p$ for the chemostat reactor in the previous question?