A rectangular reinforced concrete beam with width b $=350$ mm$,$ and effective depth of d $=500$ mm$,$ and a total depth of h $=700$ mm is reinforced with three No$.32(3$ x $819$ mm$2)$ bars. Material strength are concrete with f$$c$=35$ MPa and steel with fy $=420$ MPa. The elastic moduli of the concrete and steel are, respectively, Ec $=28,000$ MPa and Es $=200,000$ MPa. $($a$)$ Find the moment that will produce the first cracking at the bottom surface of the beam, basing your calculation on Ig$,$ the moment of inertia of the gross concrete section. $($b$)$ Repeat the calculation, using Iut, the moment of inertia of the uncracked transformed section. $($c$)$ Determine the maximum moment that can be carried without stressing the concrete beyond $0\mathrm{.}45$ f$$c or the steel beyond $0\mathrm{.}60$ fy