The rectangular beam shown below carries its own dead load plus an additional uniform dead load $($superimposed dead load$)$ of $0\mathrm{.}50ki\frac{p}{f}t,$ and a uniform live load of $1\mathrm{.}9ki\frac{p}{f}t.$ Considering the three loading cases indicated in problem $5-4$ of your textbook $($page $225)$:

a$.$ Solve the beam, draw factored bending$-$moment diagrams for the three loading cases, and superimpose them to draw a bending$-$moment envelope.

b$.$ Design a singly$-$reinforced rectangular section for the maximum positive moment between the supports $($Design case: dimensions are unknown$).$

c$.$ Starting with the beam section from part $($b$),$ design the tension reinforcement for the maximum negative moment over the roller support $($Design case: dimensions are known$).$

Note: You must use the loads and dimensions indicated here, not those in the textbook.