A factory is producing DVD and Blu Ray discs. For each case of DVDs they make $180 profit. For each case of Blu Ray they make $250. The DVDs take 3 minutes of machine time and 7 minutes of labor. The Blu Rays take 4 minutes of machine time and 12 minutes of labor. The maximum machine time available is 210 minutes. The maximum labor available is 546 minutes. How many DVDs and Blu Rays should be made to maximize profits? Find the constraints, set up a table, graph, and write a sentence.

To solve this problem, we need to set up constraints and build a table to compare the profits for each combination of DVD and Blu-ray cases.

Let's define the following variables:
- Let x be the number of cases of DVDs.
- Let y be the number of cases of Blu-ray discs.

Now, let's set up the constraints based on the available machine time and labor:
1. Machine time constraint: 3x + 4y ≤ 210 (since DVDs take 3 minutes and Blu-rays take 4 minutes each)
2. Labor constraint: 7x + 12y ≤ 546 (since DVDs require 7 minutes and Blu-rays require 12 minutes each)

Next, let's construct a table to compare the profits for each combination of DVDs and Blu-rays:
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| Cases | DVDs | Blu-rays |
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| Machine Time| 3x | 4y |
| Labor Time | 7x | 12y |
| Profit ($) | 180x | 250y |
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Now, we can plot the constraints on a graph with x-axis representing DVDs (x) and y-axis representing Blu-rays (y).

To plot each constraint, we'll assume one constraint and solve for the corresponding values:
- For the Machine Time constraint: 3x + 4y ≤ 210
Let's solve for x when y = 0: 3x ≤ 210
Dividing by 3 on both sides, we get x ≤ 70. Plotting this as a straight line on the graph.
- For the Labor constraint: 7x + 12y ≤ 546
Let's solve for x when y = 0: 7x ≤ 546
Dividing by 7 on both sides, we get x ≤ 78. Plotting this as a straight line on the graph.

The feasible region where both constraints are satisfied is the shaded area on the graph.

Lastly, to find the combination of DVDs and Blu-rays that maximizes profits, we'll need to find the corner points of the shaded region and calculate the profit for each combination. The combination with the highest profit will indicate the optimal number of DVDs and Blu-rays to produce.