$10\mathrm{.}1$ A water heating system has a fully mixed tank of capacity $400$ liters. The tank is located in a building having a temperature of $19C.$ The collector area is $5\mathrm{.}0m^2.$ The tank area$-$loss coefficient product $(UA{)}_s=0\mathrm{.}81\frac{W}{}C,F_R()=0\mathrm{.}76,$ and $F_R{U}_L=4\mathrm{.}80\frac{W}{m^2}C.$ In a particular hour when the ambient temperature is $12C,$ the radiation on the plane of the collector is $18\mathrm{.}0$ MJ$.$ The load on the system is $5\mathrm{.}20MJ.$ If the tank temperature is $43C$ at the beginning of the hour, estimate the temperature at the end of the hour.