You have been told of a mysterious stone wall $0\mathrm{.}1$ m thick with an unknown source of internal heat generation $=120,000$ W$/$m$3.$ The wall material has the properties: k$=72$ W$/$m$-$K$,$ r$=8,000$ kg$/$m$3,$ and Cp $=500$ J$/$kg$-$K$.$ The wall has been at $20\backslash$deg C$,$ but in an effort to understand the nature of the heat generation, the temperature on one side of the wall was quickly raised to $150\backslash$deg C and the other side was raised to $300\backslash$deg C$.$ As the new engineer on the team, you havent been told of the reasoning for inducing this temperature change on the walls, but you have been asked to determine the temperature profile in the wall over time. You have been told that it will be adequate for the initial evaluation of the wall to define nodes $0\mathrm{.}02$ m apart and a time step of $30$ seconds. a$)$ Set up the system of equations for an implicit solution. Develop the matrices needed to solve at each time step.b$)$ Solve for two forward time steps of $30$ seconds each using matrix algebra $($e$.$g$.,$ solve for the temperature profiles at $30$ seconds and at $60$ seconds$).$c$)$ For this assignment, calculate at least four additional time steps and plot the temperature vs distance at the various time steps and plot the temperature for the different positions as a function of time. About how long does it take to reach steady state?d$)$ Explain briefly if the trends in your plots match the trends you would have expected for heat conduction in a wall with internal heat generation before you made the calculations. Note anything that may have been unexpected at first glance