$4.$ A $10-$mm$-$thick workpiece, initially at $300$K$,$ is to be annealed at a temperature above $900$K for a period of at least $5$ minutes and then cooled. The workpiece is opaque and diffuse, and the spectral distribution of its emissivity is shown schematically below. Heating is accomplished on one side $(1\backslash$times $1$m$^2)$ of the workpiece in a large furnace whose walls and circulating air are at a temperature of $1,000$K and a convection coefficient of h$\_$L$=100$W $/$ m$^2-$K$.$ The thermophysical properties of the workpiece are $\backslash$rho $=2700$ kg $/$ m$^3,$ c$\_$p$=885$J $/$ kg$-$K$,$ and k$=165$W $/$ m$-$K$.($a$)$ Calculate the emissivity and the absorptivity of the workpiece when it is placed in the furnace at its initial temperature of $300$K$.($b$)$ Determine the net heat flux into the workpiece for this initial condition. What is the corresponding rate of change in temperature, d T $/$ d t$,$ for the workpiece?