$8\mathrm{.}19$ Consider the LRE$-$propelled, space$-$based, rocket shown in Figure $8\mathrm{.}42.$ The engine operates with a constant thrust of $\backslash (200,000\backslash$mathrm$\{$~N$\}\backslash )$ and $\backslash ($ I$\_\{\backslash$mathrm$\{$sp$\}\}\backslash )$ of $350$ s for a duration of $25$ s $.$ The vehicle mass at ignition of the engine is $2500$ kg $,$ while the payload mass $($which includes all structure forward of the fuel tank$)$ is $500$ kg $.$ The tank is fabricated from aluminum, which has

Figure $8\mathrm{.}42$ Diagram for Problem $8\mathrm{.}19.$

a Young's modulus of $6900$ MPa and a Poisson's ratio of $0\mathrm{.}3.$ Using this information, determine the minimum ullage pressure in the fuel tank to prevent buckling of the tank wall.