A building has a flat roof consisting, from inside to outside, of $38-$in$.$ gypsum wallboard, $28$ studs located on $12-$in$.$ centers, $34-$in$.$ plywood, and $38-$in$.$ built$-$up roofing. $($Note that the actual dimensions of a $28$ stud are $1\mathrm{.}5$in$.7\mathrm{.}5$in$.$ and it is made out of softwood.$)$ The $7\mathrm{.}5-$in$.-$high space between the studs is filled with loose cellulose insulation, which has a thermal conductivity of $0\mathrm{.}30$Btu$$in$./$hr$$ft$2$F$.$ If the outdoor temperature is $95$F$,$ outdoor wind speed is $7\mathrm{.}5$mph$,$ and indoor$-$air temperature is $78$F :

$($a$)$ Calculate the heat$-$loss rate per unit area using the parallel$-$path method.

$($b$)$ Calculate the heat loss rate per unit area using the isothermal$-$plane method.