At $1073$ K$,$ a $0\mathrm{.}8\%$ carbon steel is getting decarburised for $6$ hours in an atmosphere equivalent to $0\%$ carbon. Calculate the depth to which post$-$machining operations are to be carried out if the carbon content at the surface after$-$machining should not be below $0\mathrm{.}65\%.$ Given D$0=7\backslash$times $105$ m$2$ s$1$ and Q $=157$ kJ$/$mol for carbon diffusion in $\backslash$gamma $-$iron.