Problem #$1$: $(25$ points$)$ The cross section shown in Figure $1$ is subjected to an upward vertical shear force of $15$ kN $($not shown in the figure$).$ The horizontal $y-$axis and the vertical $z-$axis pass through the centroid $C$ of the cross section $($not shown on the figure$).$

a$)$ Find the vertical location of the centroid $C.$ Express your answer as a distance from the lower edge of the cross section.

b$)$ Find the moment of inertial $I_{yy}$ about the $y-$axis.

c$)$ Calculate the maximum vertical shear stress ${}_{\text{m}\text{a}\text{x}}$ on the cross section and clearly describe where it occurs.