A rectangular, tension$-$reinforced beam is to be designed for dead load of $500l\frac{b}{f}t$ plus self$-$weight and service live load of $1200l\frac{b}{f}t,$ with a $22$ ft simple span. Material strengths will be $f_y=60$ksi and $f_c^{\text{'}}=3$ksi for steel and concrete, respectively. The total beam depth must not exceed $16$ in$.$ Calculate the required beam width and tensile steel reinforcement, using a reinforcement ratio of $0\mathrm{.}60{}_{0\mathrm{.}005}.$ Use ACI load factors and strength reduction factors. The effective depth may be assumed to be $2\mathrm{.}5$ in less than the total depth. Use the following steps. $(30$ points$)$

a$.$ Calculate ${}_{0\mathrm{.}005}=0\mathrm{.}85\frac{f_c^{\text{'}}}{f_y}{}_1(\frac{3}{8})$

b$.$ Set $\frac{M_u}{b{d}^2}$ to the $\frac{M_u}{b{d}^2}$ corresponding to $0\mathrm{.}6{}_{0\mathrm{.}005}$ and calculate the required b value. Provide b larger than calculated.

c$.$ Calculate $A_{st}=0\mathrm{.}6{}_{0\mathrm{.}005}**bd$ and provide rebars to the nearest higher area

d$.$ For the proposed section, show that $M_n>M_u$