For the precast L $-$beam shown below:

$($a$)$ Compute $a,c,$ and ${}_t.$

$($b$)$ Does the beam satisfy ACI tension$-$controlled$-$failure requirement?

$($c$)$ Calculate nominal moment $M_n$ and design strength $M_n.$

Given: $f_c^{\text{'}}=5,000$; and $f_y=60,000$

$(25$ points$)$