Problem C$4.$ Find the lightest W$10$ section to carry the factored axial compression load of $495$ kips using Grade $50$ steel $(F_y=50$ ksi, $F_u=65$ ksi$).$

$($a$)$ Find the effective lengths in $x$ and $y$ directions.

$($b$)$ Select the lightest W$10$ shape.

$($c$)$ Check the design assumption and redesign if needed.