A $2-$m x $1\mathrm{.}5-$m section of wall of an industrial furnace burning natural gas is not insulated, and the temperature at the outer surface of this section is measured to be $75$C The tem$-$ perature of the furnace room is $30$C$,$ and the combined con$-$ vection and radiation heat transfer coefficient at the surface of the outer furnace is $10$ W$/$m$^2$C$.$ It is proposed to insulate this section of the furnace wall with glass wool insulation $($k$=0\mathrm{.}038$ W$/$m $$C$)$ in order to reduce the heat loss by $90$ per$-$ cent. Assuming the outer surface temperature of the metal sec$-$ tion still remains at about $80$C$,$ determine the thickness of the insulation that needs to be used. The furnace operates continuously and has an efficiency of $78$ percent. The price of the natural gas is $$0\mathrm{.}55/$therm $(1$ therm $=105,500$ kJ of energy content$).$ If the installation of the insu$-$ lation will cost $$250$ for materials and labor, determine how long it will take for the insulation to pay for itself from the en$-$ ergy it saves.