Problem $1(20$ points$)$

A cast iron $($brittle$)$ cylindrical vessel contains pressurized air. The internal diameter of the vessel is $10$ cm and its thickness is $2$ cm $.$ If the ultimate compressive and tensile strengths of the material are $750$ MPa and $215$ MPa $,$ respectively, calculate the maximum pressure the air inside the vessel can be stored at using the Coulomb$-$Mohr criterion for brittle materials. You may ignore the longitudinal stresses. Assume the factor of safety to be $1\mathrm{.}5.$