P$8\mathrm{.}5$ A beam is subjected to equal bending moments of $M_z=45$kip$*ft,$ as shown in Figure P$8\mathrm{.}5$a$.$ The cross$-$sectional dimensions $($Figure P$8\mathrm{.}5$b$)$ are $b_1=7\mathrm{.}5$ in$.,d_1=1\mathrm{.}5$ in$.,b_2=0\mathrm{.}75$ in$.,d_2=6\mathrm{.}0$ in$.,b_3=3\mathrm{.}0in.,$ and $d_3=2\mathrm{.}0in.$ Determine

$($a$)$ the centroid location, the moment of inertia about the $z$ axis, and the controlling section modulus about the $z$ axis.

$($b$)$ the bending stress at point $H.$ State whether the normal stress at $H$ is tension or compression.

$($c$)$ the bending stress at point $K.$ State whether the normal stress at $K$ is tension or compression.

$($d$)$ the maximum bending stress produced in the cross section. State whether the stress is tension or compression.