c$)$ The following equilibrium relation was obtained during treatment of an aqueous solution of a valuable solute by decolorising carbon for removal of colouring impurities

$y^{**}=8\mathrm{.}91{10}^{-5}{x}^{1\mathrm{.}66}$

where $x$ is the adsorbate concentration per kilogram of carbon, and

$Y^{**}$ is the equilibrium colour units per $kg$ of solution measured on an arbitrary scale proportional to the concentration of the coloured impurity.

It is proposed to reduce the colour of the solution to $20\%$ of its original value of $9.$ Determine the quantity of fresh carbon required per $100kg$ of the solution for a single stage operation.

$($HINT: Assume values for $x)$

$(16$ Marks$)$