By applying material balance over substrate in a continuous culture, prove that:

$P=D[Y_{\frac{x}{a}}(S_0-\frac{D{K}_s}{D_c-D})],$ and also $D_{\text{a}\text{p}\text{t}}=D_c[1-\sqrt[\phantom{2}]{\frac{K_s}{K_s+S_0}}]$

Where, $D=$ Dilution rate, time ${?}^{-1},D_c=$ Critical dilution rate, time ${?}^{-1},D_{\text{o}\text{p}\text{t}}=$ Optimum dilution rate, time ${?}^{-1},K_x=$ Saturation constant, concentration unit $(\frac{\text{m}\text{a}\text{s}\text{s}}{\text{v}\text{o}\text{l}\text{u}\text{m}\text{e}}),P=$ Productivity, $\frac{\text{m}\text{a}\text{s}\text{s}}{\text{t}\text{i}\text{m}\text{e}\text{.}\text{v}\text{o}\text{l}\text{u}\text{m}\text{e}},S_0=$ Substrate initial concentration, concentration unit $(\frac{\text{m}\text{a}\text{s}\text{s}}{\text{v}\text{o}\text{l}\text{u}\text{m}\text{e}}),Y_{\frac{x}{4}}=$ Biomass yield coefficient