Aiba et al$.(1968)$ reported the results of a chemostat study on the growth of a specific strain of baker's yeat as shown in the following table. The inlet stream of the chemostat did not contain any cells or products:

$\backslash$table$[[$Dilution Rate,,Inlet concentration,Steady State concentration $($g$/$L$)],[$D$,1/$hr$,($g$/$L$),$ Glucose, Cs$,$Glucose, Cs$,$Ethanol, Cp$,$Cells, Cx$],[0\mathrm{.}084,21\mathrm{.}5,0\mathrm{.}054,7\mathrm{.}97,2\mathrm{.}00],[0\mathrm{.}100,10\mathrm{.}9,0\mathrm{.}079,4\mathrm{.}70,1\mathrm{.}20],[0\mathrm{.}160,21\mathrm{.}2,0\mathrm{.}138,8\mathrm{.}57,2\mathrm{.}40],[0\mathrm{.}198,20\mathrm{.}7,0\mathrm{.}186,8\mathrm{.}44,2\mathrm{.}33],[0\mathrm{.}242,10\mathrm{.}8,0\mathrm{.}226,4\mathrm{.}51,1\mathrm{.}25]]$

$($a$)$ Find the rate equation for cell growth in the form of $\frac{{}_m**S}{K_S+S}**x.$ Determine ${}_m$ and $K_S$ from the data and then write the final form of the equation.