Problem $6(25$ points$)$ A box beam composed of two welded channels is shown in Fig. $1.$ To strengthen the joint a plate is welded to the bottom of the box beam as shown in Fig. $2.$ The channels are each C$8$x$11\mathrm{.}5$ shapes. The plate is $0\mathrm{.}5$ inches by $4\mathrm{.}0$ inches in cross$-$section.

$($a$)$ What is the value of the $\backslash ($ y $\backslash )$ component of the overall centroid for the beam in Figure $2($measured from the bottom of the plate$)?$

$($b$)$ What is the moment of inertia about the x axis of the overall centroid for the same beam?

$17$