Q$.4($a$)$ Figure Q$\mathrm{.}4($a$)$ shows the cross$-$section of the U$-$section with centroidal axes at $x-x$ and $Y-Y.$

$($i$)$ Calculate the distance $x_c$ and $Y_c$ measured from the point $O$ of the section.

$(4$ marks$)$

$($ii$)$ Calculate the moment of inertia $(I_{})$ of the section about $x-x$ axis. $(5$ marks$)$

$($iii$)$ Calculate the moment of inertia $(I_{yy})$ of the section about $Y-Y$ axis.

$(5$ marks$)$

Figure O$4($a$)$

$($b$)$ A cantilever beam is subjected to uniformly distributed loads $($UDL$)$ as shown in Figure Q$\mathrm{.}4($b$).$ Determine the maximum deflection and the maximum slope of the beam due to the applied loads. Given that the flexural rigidity $(EI)$ of the beam is $90,000kN{m}^2.$

$(11$ marks$)$