A mixed$-$growth$-$associated product formation for a chemostat culture is given by the

following equation: $d\frac{P}{d}t=({}_g+)x.$

The kinetic parameter values are ${}_m=0\mathrm{.}7h^{-1},K_s=20m\frac{g}{L},Y_{\frac{x}{s}}=0\mathrm{.}5g$ dw $\frac{?}{g}$ substrate,

$Y_{\frac{?}{PI}x}=0\mathrm{.}15g\frac{P}{g}dw,=0\mathrm{.}1,=0\mathrm{.}02h^{-1},$ and $S_0=1\frac{g}{L}.$

a$.$ Determine the optimal dilution rate maximizing the productivity of produt formation

$($DP$).$

b$.$ Determine the optimal dilution rate maximizing the productivity of cell formation $($DX$).$