A propeller shaft of length L, loaded in torsion, must support a specified torque T without failing and be as cheap as possible. Section shape is a variable and "failure" means the first onset of plasticity. The table below summarizes the requirements. Function Constraints Objective Free variables Cheap shaft Specified failure torque T Length L specified Minimize material cost C Choice of material Section shape and scale (a) Derive the material index. (b) The table below shows the dimensions of 5 candidate materials: High carbon steel, low alloy steel, titanium alloys and CFRP (carbon fiber reinforced epoxy). Use the index you derived to rank these candidate materials for this application using data from Appendix A. Material High carbon steel Titanium alloys Aluminum alloy Magnesium alloy CFRP (carbon fiber reinforced epoxy) R (mm) t (mm) 70 50 90 80 100 8 6 10 10 7