Problem #$3$: $(20$ points$)$ Consider a cantilever beam with a T$-$shaped cross section subjected to a

bending moment M of magnitude $200$N$-$m applied at its free end, as shown in Figure $3.($On the

cross section, use a y$-$z coordinate system with origin at the centroid C of the cross section, in

which the horizontal z$-$direction is positive to the left through C and the vertical y$-$direction is

positive upward through C $.)$ The dimensions of the cross section are given on the figure.

a$)$ Find the vertical location of the centroid C of the cross section. Calculate your result as a

distance from the bottom edge of the cross section. If you treat the cross section as a

composite area, draw a picture that clearly identifies and labels the subareas.

b$)$ Find the moment of inertia I$\_($z$)$ of the cross section about the z$-$axis $($which points horizontally

to the left through C $).$ If you treat the cross section as a composite area, draw a picture that

clearly identifies and labels the subareas.

c$)$ Calculate the compressive bending stress sigma$\_("$comp $")$ of greatest magnitude on the cross section,

and clearly describe where it occurs on the cross section.

d$)$ Calculate the tensile bending stress sigma$\_("$tens $")$ of greatest magnitude on the cross section, and

clearly describe where it occurs on the cross section. Please show all the details and please check all answers, I will give a LIKE!