$(10$ pts$.)$ You are tasked with designing a cylindrical pressure vessel for high$-$pressure applications. The internal pressure is $500$ MPa $,$ and the yield strength of the material is $400$ MPa $.$ The vessel's radius is $800$ cm $,$ and its wall thickness must be optimized. Using the Von Mises yield criterion, derive an expression for the minimum required wall thickness to prevent yielding, considering both hoop and longitudinal stresses.