For the angle shape shown, calculate

$($a$.)$ the moments of inertia $I_x$ and $d_x$ about the $x$ and $y$ centroidal axes, respectively.

$($b$.)$ the product of inertia

$($c$.)$ the principal moments of inertia $I_{\text{m}\text{a}\text{x}}$ and $I_{\text{m}\text{i}\text{n}\text{.}}.$

$($d$.)$ the angle ${}_p.$ Sketch the axis that corresponds to $I_{\text{m}\text{a}\text{x}}$ on the cross section and show the orientation of angle ${}_p$ on the sketch.