7.5 You have been asked to prepare an outline design for the pressure hull of a deep-sea submersible vehicle capable of descending to the bottom of the Mariana Trench in

the Pacific Ocean-the deepest water on the planet (Challenger Deep, 1120 57 N 142 13 00 E, 11,000 m depth). The external pressure at this depth is approximately 100 MN m^-2, and the design pressure is to be taken as 200 MN m?.The pressure hull is to have the form of a thin-walled sphere with a specified radius r of 1 m and a uniform thickness t. The sphere can fail by external-pressure buckling at a pressure p, given by P_{b}=0.3 E\left(\frac{t}{I}\right)^{2} where E is Young's modulus. The basic design requirement is that the pressure hull shall have the minimum possible mass compatible with surviving the design pressure. By eliminating t from the equations, show that the minimum mass of the hull is given by the expression M_{\mathrm{b}}=22.9 r^{3} p_{\mathrm{b}}^{0.5}\left(\frac{\rho^{2}}{E}\right)^{1 / 2} Hence obtain a materials index to meet the design requirement for the failure mechanism. [You may assume that the surface area of the sphere is 4nr^2].