12.1 Referring to Example 7.5, the sphere can fail by yield or compressive fracture at a pressure p1 given by p_{\mathrm{A}}=2 \sigma_{\mathrm{t}}\left(\frac{t}{I}\right) where o is the yield stress or the compressive fracture stress as appropriate.The basic design requirement is that the pressure hull has 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{f}}=2 \pi r^{3} \mathrm{p}\left(\frac{\rho}{\sigma_{\mathrm{i}}}\right) Hence obtain a merit index to meet the design requirement for the failure mechanism. [You may assume that the surface area of the sphere is 4nr^2.]

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