A house is $24\text{'}$ wide, $48\text{'}$ long, and $17\text{'}$ high. The $2$x$6$ frame wall has studs $24"$ on center and has $51/2"$ fiberglass insulation. The interior is $1/2"$ gypsum board. The outer sheathing is $1/2"$ plywood with $3/4"$ barnboard siding. The double glazed windows occupy $15\%$ of the wall area of the house and are $3/16"$ glass with $1/2"$ air gap. The roof pitch is $4$:$12,$ and the rafters are $2$x$10$ with $9"$ of fiberglass insulation. The $5/8"$ plywood on these rafters is topped with asphalt shingles. The upper level has cathedral ceilings, so the finished ceiling is attached directly to the rafters. That finished ceiling is $3/4"$ tongue & groove boards. The interior temperature is $70\backslash$deg F$,$ and the exterior temperature is $30\backslash$deg F$.$ Assume the basement losses are negligible. Assume $7\mathrm{.}5$ mph wind speed. Assume the air gap in the windows is $1/2"$

$($a$)$ Determine the average conductance of the sidewall, taking into account fenestration.

$($b$)$ Determine the R factor for the roof.

$($c$)$ Find the total energy losses from the house in BTU$/$hr$.$