Chemical Reactor Engineering Problem!

Please answer this question as soon as possible with specific explanation.

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The production of a product $P$ from a particular bacteria follows the Monod growth law, with ${}_{\text{m}\text{a}\text{x}}=1h^{-1},$

$K_s=0\mathrm{.}25\frac{g}{d}{m}^3$ and $Y_{\frac{c}{s}}=0\mathrm{.}5\frac{g}{g}.$

$($a$)$ If the reaction is first tested in a batch reactor with the initial cell concentration of $C_0=0\mathrm{.}1\frac{g}{d}{m}^3$ and

substrate concentration $C_{s0}=20\frac{g}{d}{m}^3.$ Plot $C_s$ and $C_c$ as a function of time.

$($b$)$ The reaction is now scaled up in a CSTR with $C_{s0}=20\frac{g}{d}{m}^3$ and $C_{c0}=0.$ What is the dilution rate

when wash$-$out occurs and also when maximum production of cells occurs?

$($c$)$ Repeat part $($b$)$ if we include cell death with $k_d=0\mathrm{.}02h^{-1}$ and substrate maintenance with $m=0\mathrm{.}2$

$\frac{g}{h}\frac{?}{d{m}^3}.$