Muir manufacturing produces two popular grades of commercial carpeting among its many other products. in the coming production period, muir needs to decide how many rolls of each grade should be produced in order to maximize profit. each roll of grade x carpet uses 50 units of synthetic fiber, requires 25 hours of production time, and needs 20 units of foam backing

Part A:

Given that the decision that Muir company needs to make is how many rolls of each grade of carpet it should produce in order to maximize profit.

Therefore, the decision variables are the number of grade X carpets Muir company should produce which we will denote as x, and the number of grade Y carpets they should produce, which we will denote as y.

Part B:

From the question, we are interested in maximizing profit.

Given that the profit per roll of Grade X carpet is $200 and the profit per roll of Grade Y carpet is $160.

Therefore, the objective function is given by:

P = 200x + 160y

Part C:

Given that each roll of Grade X carpet uses 50 units of synthetic fiber, each roll of Grade Y carpet uses 40 units of synthetic fiber and Muir has 3000 units of synthetic fiber available for use.

Thus, 50x + 40y ≤ 3000

Given that each roll of Grade X carpet requires 25 hours of production time, each roll of Grade Y carpet requires 28 hours of production time and workers have been scheduled to provide at least 1800 hours of production time.

Thus, 25x + 28y ≥ 1800

Given that each roll of Grade X carpet needs 20 units of foam backing, each roll of Grade Y carpet needs 15 units of foam backing and the company has 1500 units of foam backing available for use.

Thus, 20x + 15y ≤ 1500

Therefore, the constraints are:

50x + 40y ≤ 3000
25x + 28y ≥ 1800
20x + 15y ≤ 1500

Part D:

From the graph of the constraints, we can see that the corner points bounded by the lines of the graph of the three constraints are: (30, 37.5), (0, 64) and (0, 75)

Checking the corner points with the objective function, we have:

For (30, 37.5): P = 200(30) + 160(37.5) = 6,000 + 6,000 = 12,000
For (0, 64): P = 200(0) + 160(64) = 10,240
For (0, 75): P = 200(0) + 160(75) = 12,000

Therefore, to maximize profit, Muir should produce 30 grade X carpets and 37.5 grade Y carpets or no grade X carpet and 75 grade y carpet.