$1.(16$ marks$)$ A simply supported beam is subjected to the loads shown below in Fig. $($a$)$ and its cross section is demonstrated in Fig. $($b$).$ Determine the required cross$-$sectional dimension, $\backslash ($ a $\backslash ),$ if the maximum allowable tensile stress is $110$ MPa and the maximum allowable compressive stress is $100$ MPa for the material.