For each of the following linear programming models, give your recommendation on which is the more efficient way (probably) to obtain an optimal solution: by applying the simplex method directly to this primal problem or by applying the simplex method directly to the dual problem instead. Explain. (a) Maximize Z= 10x1 - 4x2 + 7x3, subject to 3x - x2 + 2xz < 25 X1 - 2x2 + 3x3 = 25 5x + x2 + 2x3 = 40 x1 + x2 + x3 = 90 2x1 - x2 + x3 = 20 and x, 20, 20, x2 > 0. (b) Maximize Z = 2xy + 5x2 + 3x3 + 4X4 + X5, subject to x1 + 3x2 + 2x3 + 3x4 + x3 = 6 4x1 + 6x2 + 5xz + 7x4 + x3 = 15 and x; 20, for j = 1, 2, 3, 4, 5.

Anew firm is developing its business plan. it will require $615,000 of assets, and it projects $450,000 of sales and $355,000 of operating costs for the first year. management is reasonably sure of these numbers because of contracts with its customers and suppliers. it can borrow at a rate of 7.5%, but the bank requires it to have a tie of at least 4.0, and if the tie falls below this level the bank will call in the loan and the firm will go bankrupt. what is the maximum debt ratio the firm can use? (hint: find the maximum dollars of interest, then the debt that produces that interest, and then the related debt ratio.)a. 41.94%b. 44.15%c. 46.47%d. 48.92%e. 51.49%