$5\mathrm{.}31.$ Consider the heating of a house by a furnace, which serves as a heat$-$source reservoir at a high temperature $T_F.$ The house acts as a heat$-$sink reservoir at temperature $T,$ and heat $|Q|$ must be added to the house during a particular time interval to maintain this temperature. Heat $|Q|$ can of course be transferred directly from the furnace to the house, as is the usual practice. However, a third heat reservoir is readily available, namely, the surroundings at temperature $T_{},$ which can serve as another heat source, thus reducing the amount of heat required from the furnace. Given that $T_F=810K,T=295K,T_{}=265K,$ and $|Q|=1000kJ,$ determine the minimum amount of heat $|Q_F|$ which must be extracted from the heat$-$source reservoir $($furnace$)$ at $T_F.$ No other sources of energy are available.