A hot brass plate $($p $=8530$ kg$/$m$^3,$ cp $=380$ J$/$kg$-$K$,$ k $=110$ W$/$m$-$K$,$ and a $=33\mathrm{.}9$ x $10-6$ m$^2/$s$)$ with an initial temperature of $650$C and a thickness of $10$ cm is insulated on the lower surface, and its upper surface is cooled by an impinging air jet with a temperature of $15$C$,$ with a convection heat transfer coefficient of $220$ W$/$m$^2-$K$.$ Find the mid$-$plane temperature of the brass plate $3$ minutes after it starts cooling.