Problem $3(15$ points$).$ A brittle material has ultimate strengths of $40$ ksi and $90$ ksi in tension and compression, respectively. Find the factors of safety guarding against failure for the following plane stress states using BCM and MM theories.

$($a$){}_x=12$ksi,${}_y=0,{}_{xy}=-8$ksi

$($b$){}_x=-6$ksi,${}_y=-10$ksi,${}_{xy}=-5$ksi

$($c$){}_x=-12$ksi,${}_y=0,{}_{xy}=8$ksi