A manufacturer produces three models of bicycles. The times (in hours) required for assembling, painting, and packaging each model are shown in the table below: Model A Model B Model Assembling 2 2.5 3 Painting 1.5 2 1Packaging 1 0.75 1.25 The total time available for assembling is 4006 hours, for painting is 2495 hours, and for packaging is 1500 hours. The profits per unit are $45 for Model A. $50 for Model B and $55 for Model C. What is the optimal production level for each model, and what is the optimal profit? A) Set up the objective function. B) Set up the constraints in inequality form. C) Rewrite your answer to (b) as equations with slack variables. D) Write the initial tableau for solution via the simplex method.

we have to maximize the following equation:

45A + 50B + 55C

where:

A = number of model A bicycles produced

B = number of model B bicycles produced

C = number of model C bicycles produced

the constraints are:

2A + 2.5B + 3C ≤ 4006 (assembly constraint)

A + 0.5B + 2C ≤ 2495 (painting constraint)

A + 0.75B + 1.25C ≤ 1500 (packaging constraint)

A,B,C ≥ 0

using solver, the optimal solution is: 745A + 1006B = $83,825

using slack variables:

2A + 2.5B + 3C + S1 = 4006 (assembly constraint)

A + 0.5B + 2C + S2 = 2495 (painting constraint)

A + 0.75B + 1.25C + S3 = 1500 (packaging constraint)

A,B,C,S ≥ 0

slack variable tableau:

A         B         C           S1          S2          S3           Z            B

2         2.5       3           1             0            0            0            4006

1          0.5       2          0             1            0             0            2495

1          0.75     1.25      0             0           1              0            1500

-45      -50      -55       0             0           0              1            0

the answer is c: divide the arc's degree measure by 360°, then multiply by the circumference of the circle.