3 use the graphical method for the following problem

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3. ( Use the graphical method for the following problem ): A technology company manufac- tures televisions and soundbars. The following table gives the labor requirements and profits per automobile: Televisions Soundbars Assemble 10 labor-hours 8 labor-hours Load and Transport 2 labor-hours 1 labor-hours Profit $150 $100 Each day, 360 labor-hours are available to assemble these products and 60 labor-hours are avail- able to load and trasport them. The manufacturer wants to determine how many televisions and soundbars to produce of each type in order to maximize profit under these constraints. (a) [5 points] Define variables for modeling this scenario, and write down the objective function.
(b) Identify the constraints of the problem.
Assembly: 10 x + 8 y 360 , Load and Transport: 2 x + y 60 , We also have our physical constraints (no negative items can be produced): x 0 , y 0 . The complete system of constraints is given by 10 x + 8 y 360 2 x + y 60 x 0 , y 0 (c) List the corner points of the feasible region.
Thus the corner points are (0 , 0) , (0 , 45) , (20 , 20) , (30 , 0) . (d) [5 points] How many televisions and soundbars should be produced each day to maximize the total profit?
4. A colleague approaches you with the following simplex tableau: x y z s t u p 0 5 0 1 4 0 0 72 1 0 0 0 3 1 0 60 1 1 1 0 1 0 0 22 3 -2 0 0 -4 0 1 140 (a) [10 points] Your colleague does not remember which variables to list in the first column of the tableau. Write down these variables, and give their values.

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