$(60$ points$)$ For the section shown below, find:

a$.$ The centroid measured from point A$.($Result check: $\stackrel{}{x}=\frac{597}{41}cm\frac{,}{b}ar(y)=12cm)$

b$.$ The moments of inertia and the product of inertia about the $x^{\text{'}}-$ and $y^{\text{'}}-$axes with the origin in the centroid of the section. $($Result check: $\frac{?}{\text{b}\text{a}\text{r}}(I{)}_{x^{\text{'}}}=29,353\mathrm{.}5c{m}^4,\frac{?}{\text{b}\text{a}\text{r}}(I{)}_{y^{\text{'}}}=32,909\mathrm{.}4c{m}^4\frac{,}{b}ar(I{)}_{x\text{'}{y}^{\text{'}}}=0c{m}^4)$

c$.$ The radii of gyration. $($Result check: $k_{x\text{'}}=8\mathrm{.}92cm,k_{y^{\text{'}}}=9\mathrm{.}44cm)$

Use moment of inertia and product of inertia formulas of axes through the centroid