The railroad I$-$beam has dimensions as shown below. Each of the three segments of the Ibeam have a length and width as indicated in the diagram. In reviewing these old designs, you have been asked to evaluate the area moment of inertia of the I$-$beam in several orientations.

a$)$ Calculate area moment of inertia $-$ Ix and ly $-$ for the I$-$beam shown. Note that the x $-$axis and $y-$axis pass through the centroid of the I$-$beam.

b$)$ What is the product of inertia lxy$?$ Show calculations or provide a justification.

c$)$ Calculate area moment of inertia of the inclined axis $($lu$)$ in the direction parallel to the force.

d$)$ Calculate area moment of inertia of the inclined axis $($lv $-$ not drawn$)$ in the direction perpendicular to the force $($but passing through the centroid$).$

$20$ POINTS