Ethanol formation from glucose is accomplished in a batch culture of Saccharomyces cerevisiae, and the following data were obtained.

$\backslash$table$[[$Time $($h$),$Glucose $($S$),$ g$/$L$,$Biomass $(x),\frac{g}{L},$Ethanol $($P$),$ g$/$L$],[0,100,0\mathrm{.}5,0\mathrm{.}0],[2,95,1\mathrm{.}0,2\mathrm{.}5],[5,85,2\mathrm{.}1,7\mathrm{.}5],[10,58,4\mathrm{.}8,20\mathrm{.}0],[15,30,7\mathrm{.}7,34\mathrm{.}0],[20,12,9\mathrm{.}6,43\mathrm{.}0],[25,5,10\mathrm{.}4,47\mathrm{.}5],[30,2,10\mathrm{.}7,49\mathrm{.}0]]$

a$.$ By fitting the biomass data to the logistic equation, determine the carrying$-$capacity coefficient $k.$

b$.$ Determine yield coefficients $Y_{\frac{P}{S}}$ and $Y_{\frac{x}{S}}.$