a. A material for making a shelf of length L and widthb that carries a load of books is required. The deflection at mid-span should not exceed a specified value S max when the bookcase is fully loaded with average-sized books. Assume that the thicknesst is variable. If the goal is to minimize the mass of the bookshelf, translate the problem into: i. Function ii. Constraints iii. Objectives iv. Free variables b. Repeat part a of this problem for the goal of minimizing the cost of the bookshelf Model the book shelf as a simply supported beam supporting a distributed load of w per unit length. Derive the material index for i. minimizing mass ii. minimizing cost d. Calculate the appropriate materials index for 5 materials of your choosing, and discuss the results, for the following goals: i. minimize mass ii. minimize cost c. a. A material for making a shelf of length L and widthb that carries a load of books is required. The deflection at mid-span should not exceed a specified value S max when the bookcase is fully loaded with average-sized books. Assume that the thicknesst is variable. If the goal is to minimize the mass of the bookshelf, translate the problem into: i. Function ii. Constraints iii. Objectives iv. Free variables b. Repeat part a of this problem for the goal of minimizing the cost of the bookshelf Model the book shelf as a simply supported beam supporting a distributed load of w per unit length. Derive the material index for i. minimizing mass ii. minimizing cost d. Calculate the appropriate materials index for 5 materials of your choosing, and discuss the results, for the following goals: i. minimize mass ii. minimize cost c