Two I$-$beam designs $($shown below$)$ have been submitted for a project, and your engineering firm is tasked with checking whether they meet the design specifications for area moment of inertia.

a$)$ Calculate the location of the centroid $(\stackrel{}{x}\frac{,}{b}ar(y))$ for the composite shapes shown in design A and in design $B.$

b$)$ Calculate the area moment of inertia $((x))$ for design A and design B$.$ Make sure to use the centroidal axis $\stackrel{}{x}.$

c$)$ Calculate the area moment of inertia $($ly$)$ for design A and design B$.$ Make sure to use the centroidal axis $\frac{?}{\text{b}\text{a}\text{r}}(y).$

d$)$ Calculate polar moment of inertia $(Jz)$ for design A and design $B($through each centroid$).$

$30$ POINTS