Problem $2.$ A $200$ mm by $200$ mm square plate is placed over a furnace. The plate has mass of $2$ kg $,$ a specific heat of $900\frac{J}{k}gK,$ and an emissivity of $0\mathrm{.}8.$ At some moment in time it is found to have a temperature of $400$ K $.$ The wall of the furnace are perfectly insulated. The bottom of the furnance is a blackbody maintained at $1000$ K $.$ The top of the plate is exposed to air and surroundings at $300$ K with a convective coefficient of $25\frac{W}{m^2}K.$

Find:

a$)$ The net radiative flux to the bottom of the plate.

b$)$ The rate of temperature change of the plate $\frac{dT}{dt}$ at this moment.