Given the rectangular simply supported beam and the following information: width$=12$ in$,$ depth of $18$ in span $=30ft,$ live load $=0\mathrm{.}64klf,$ self$-$weight $W_G=0\mathrm{.}225klf\frac{,}{b}ar({}_{tl})=-189\frac{,}{b}ar({}_{cl})=2400\frac{,}{b}ar({}_{ts})=0\frac{,}{b}ar({}_{cs})=3000,=0\mathrm{.}8,(e_0{)}_{mp}=7in.$

a$)$ Determine the magnitude and eccentricity of the minimum prestressing force at midspan and $($:$e_0\}.$

b$)$ Build graphically the feasible domain.