A company uses three machines, I, II, and III to produce items X, Y, and Z. The production of each item involves the use of more than one machine. It takes 2 minutes on I and 4 minutes on II to manufacture a single X. It takes 3 minutes on I and 6 minutes on III to manufacture a Y. It takes 1 minute on I, 2 minutes on II, and 3 minutes on III to manufacture a Z. The total time available on each machine per day is 6 hours. The profits are $10, $8, and $12 on each of X, Y, and Z, respectively. How should the company allocate the production times on the machines in order to maximize total profit?

L1 - 3 2 1 2 3 2 − 2 2x(1 − x ) 3+ 3x (1 − x ) 3 ex. a) Simplify: 2 2 (1 − x ) 3 2 1/3 2 3 2 −2/3 2x(1 − x ) + 3x (1 − x ) b) Solve for x: 2 2/3 =0 (1 − x )