The pressure in a small pressure vessel varies continually between the extremes of $50$ psig and $350$ psig $(340$ kPa and $2400$ kPa $).$ The vessel is closed by a $\frac{1}{2}$ in $(12mm)$ plate. The plate is attached to the vessel flange with six $\frac{3}{8}-24$ UNF bolts evenly spaced around a $9^{\frac{1}{2}}$ in $(240$ mm $)$ circle. Each bolt is tightened to an initial preload of $3700lbf(16\mathrm{.}4kN).$ The bolts and nuts are constructed of cold$-$rolled steel with a $90$ksi$(620$MPa$)$ yield strength and $110$ ksi $(760$ MPa$)$ ultimate strength. The plate, flange, and vessel are constructed of steel with a $)$ yield strength and a $50$ ksi $(345$ MPa $)$ ultimate strength. The ressel is intended to be used indefinitely. Neglect the effects of the gasket $($not shown$).$ Neglect bending of the plate. Using a thread stress concentration factor of $2$ and appropriate endurance strength derating factors, what is the factor of safety?

Note: Analyze safety for the bolt only