3. Given that the optimal solution of the following linear programming problem is x = 15 and y = 10, State the problem in standard form and do a constraint analysis for the optimal solution. Maximize 50x + 40y Subject to 4x + 2y 80 3x + 5y 60 y 10 x, y > 0

4. A company manufactures two kinds of pinball machines, each requiring a different manufacturing technique. Each Super Ball machine requires 20 hours of labor, 7 hours of testing, and yields a profit of \$300. Each Silver Ball machine requires 12 hours of labor, 8 hours of testing, and yields a profit of \$200. There are 2000 hours of labor and 1000 hours of testing available. The company has made contracts with the retailers to provide at least 60 Super Ball machines and at least 50 Silver Ball machines. The manufacturer wants to determine how many of each kind of pinball machines to manufacture. The objective is to maximize the total profit. Formulate a linear programming model for the above situation by determining (a) The decision variables. (b) The objective function. (c) All the constraints. Note: Do NOT solve the problem after formulating.

5. Charm City Foods manufactures a snack bar by blending two ingredients: a nut mix and a granola mix. Information about the two ingredients (per ounce) is shown below. Ingredient Cost In Dollars Fat Grams Protein Grams Calories Nut Mix 0.80 22 6 600 Granola Mix 0.40 2 2 50 The company needs to develop a linear programming model whose solution would tell them how many ounces of each mix to put into the snack bar. The blend should contain no more than 1000 calories, at least 10 grams of protein, and no more than 30 grams of fat. In addition, at least one ounce of nut mix must be included in the blend. Formulate a linear programming model for the above situation by determining (a) The decision variables. (b) The objective function. (c) All the constraints. Note: Do NOT solve the problem after formulating.

6. Determine whether the following linear programming problem is infeasible, unbounded, or has multiple optimal solutions. Draw a graph to find the feasible region (if it exists) and explain your conclusion. Maximize 20xl + 25×2 Subject to: 2xl + x2 10 x2 0