$12\mathrm{.}0(20$ pts$)$ Determine the time in hours required to increase the carbon content to $1\mathrm{.}28wt\%$ at a depth of $4\mathrm{.}1mm$ in a component made of AISI $1144$ steel $(0\mathrm{.}44\%$ carbon$).$ Assume the carbon concentration at the surface is $2\mathrm{.}82wt\%$ and $($ficticious$)$ diffusivity constant at $927C$ is $1\mathrm{.}54E-11\frac{m^2}{s}.$