$1.$ Determine if the $\backslash (4\backslash$times $10\backslash )$ roof purlin below is adequate to support the given loading using $($a$)$ ASD and $($b$)$ LRFD$.$

$($i$)$ Consider only bending and that the purlin is simply supported.

$($ii$)$ The distributed dead $($D$)$ and roof live $($LR$)$ load on the purlin are $\backslash (50\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}\backslash )$ and $\backslash (70\backslash$mathrm$\{$lb$\}/\backslash$mathrm$\{$ft$\}\backslash )$ respectively and there is no other loading.

$($iii$)$ The species group is Hem$-$Fir and the grade is No$.1\backslash$& Btr$.$

$($iv$)$ Use ASD load combination D$+$LR and LRFD load combination $1\mathrm{.}2$D$+1\mathrm{.}6$LR$+($L or $0\mathrm{.}5$W$).$

$($v$)$ Consider wet$-$service and normal temperature conditions and that incising is needed for wood treatment.

$($vi$)$ Consider lateral$-$torsional buckling $($Hint: there may not be a need for calculations$).$

Figure $1.$ Roof plan of a single$-$story single family wood house