$(25$pts$)$ A T$-$beam is made by nailing two $2$inx$12$in pieces of wood together with rows of a single nail each as shown. Take the bottom center as $x=0,y=0$ for the coordinate axis. The shear force $V=40llbf.$

a$.$ Show that the centroid is $xc=0in,$ yc $=9\mathrm{.}5$ in$.$

b$.$ Show that the $2^{nd}$ moment of the area is Ixc$=884$in ${?}^4.$

c$.$ Show that $\frac{?}{\text{b}\text{a}\text{r}}(y^{\text{'}})=3\mathrm{.}5$ in and $A^{\text{'}}=24i{n}^2.$

d$.$ If the spacing $s=4$ in$,$ what is the shear force on each nail?

e$.$ If the maximum shear force on each nail is $800lbf,$ what is the minimum spacing $s?$