A steel sheet with $4$ mm thickness is held at $760$C where the carbon concentration is kept constant at $3$ at$\%$ at one side and at zero at the other one. Consider steady state conditions, determine and plot the carbon concentration profile across the sheet and specify the location and concentration between austenite and ferrite phases assuming

local equilibrium. Base your analysis on a linear approximation of the phase diagram by taking $912\backslash$deg C as the transformation temperature in pure Fe and carbon equilibrium concentrations at the eutectoid temperature of $727\backslash$deg C to be $0\mathrm{.}022$ wt$\%$ and $0\mathrm{.}76$ wt$\%$ in ferrite and austenite, respectively. The diffusion coefficients of carbon at $760\backslash$deg C are approximately $5\backslash$times $10-6$ cm$2/$s in ferrite and $5\backslash$times $10-8$ cm$2/$s in austenite. The molar mass of Fe is $56$ g$/$mol and that of C is $12$ g$/$mol$.($Note: Use at$\%$ for concentrations involved in Fick$$s first law and make the conversion from wt$\%$ to at$\%$ if needed$)$