$4.$ A coupling $($see figure $4)$ is used to join two solid steel shafts. The coupling is integrally forged with the shaft. The diameter of the shaft is $\backslash ($ d$\_\{0\}=80\backslash$mathrm$\{$~mm$\}\backslash )$ and $\backslash (\backslash$mathrm$\{$D$\}\_\{\backslash$mathrm$\{$BC$\}\}=\backslash )120$ mm $,$ the flange has a thickness of $\backslash ($ t$\_\{$F L$\}=10\backslash$mathrm$\{$~mm$\}\backslash ).$ There are $6$ bolts used to connect the two flanges. If the ultimate shear stress for the bolts, shaft and flanges is $200$ MPa and the Factor of Safety is $2\mathrm{.}15,$ determine:

a$.)$ The bold diameter $($in mm $)$ required to develop the full shear capacity of the shaft. $(10$ marks$)$

b$.)$ The bearing stresses on each bolt. $(5$ marks$)$

c$.)$ If the shaft rotates at $100$ rpm $,$ determine the horsepower the shaft can transmit. $(10$ marks$)$