Figure $8\mathrm{.}41$ Diagram for Problem $8\mathrm{.}16.$

$8\mathrm{.}16$ The bipropellant launch vehicle shown in Figure $8\mathrm{.}41$ is to be used to accelerate a $\backslash (30,000\backslash$mathrm$\{$~kg$\}\backslash )$ payload $($containing an upper$-$stage propulsion system$).$ The vehicle has a gross liftoff weight $($mass$)$ of $\backslash (400,000\backslash$mathrm$\{$~kg$\}\backslash )$ and the propulsion system has an overall mass fraction of $0\mathrm{.}80.$ The liquid rocket engine provides a constant thrust of $4\mathrm{.}5$ mN $,$ while the pressurization system maintains a constant ullage pressure of one atmosphere $($absolute pressure$).$ Assuming the fuel is kerosene $\backslash (\backslash$left$(\backslash$rho$=800\backslash$mathrm$\{$~kg$\}/\backslash$mathrm$\{$m$\}^\{3\}\backslash$right$)\backslash ),$ and that the engine burns out in vacuum conditions, determine if the tank thickness of $6$ mm is adequate to avoid buckling of the tank wall at burnout of the rocket. The tank is fabricated from aluminum which has a Young's modulus of $6900$ MPa and a Poisson's ratio of $0\mathrm{.}3.$