If a profit of $\$20$ is realized for each regular gadget and $\${}30$ for a premium gadget, how many of each should be manufactured to maximize profit each week? Enter the ordered pair $(x,y).$
Hint(s):
Plot the inequalities from part (a), and find the vertices of the region formed by all the inequalities.

To maximize the profit when manufacturing regular and premium gadgets, we need to first set up the objective function and constraints, then solve the problem using linear programming.

### Objective Function:
The profit, $ P $, is given by:
\[ P = 20x + 30y \]
where $ x $ is the number of regular gadgets and $ y $ is the number of premium gadgets.

### Constraints:
Let's assume that the constraints are provided as follows:
1. There is a maximum number of regular gadgets that can be produced.
2. There is a maximum number of premium gadgets that can be produced.
3. There is a combined maximum for both types of gadgets.
4. Non-negativity constraints: $ x \geq 0 $ and $ y \geq 0 $.

For simplicity, let’s assume the following constraints for this example:
1. $ x \leq 40 $ (cannot produce more than 40 regular gadgets)
2. $ y \leq 30 $ (cannot produce more than 30 premium gadgets)
3. $ x + y \leq 50 $ (cannot produce more than 50 gadgets in total)

### Plotting the Inequalities:
1. $ x \leq 40 $
2. $ y \leq 30 $
3. $ x + y \leq 50 $
4. $ x \geq 0 $
5. $ y \geq 0 $

These constraints form a feasible region on the coordinate plane, which is a polygon bounded by the lines and the axes.

### Finding the Vertices:
The vertices of the feasible region are the points of intersection of the boundary lines. Let’s find these vertices:

1. Intersection of $ x \leq 40 $ and $ y \leq 30 $:
   \[ (40, 30) \]

2. Intersection of $ x \leq 40 $ and $ x + y \leq 50 $:
   \[ x + y = 50 \]
   \[ x = 40 \]
   \[ 40 + y = 50 \implies y = 10 \]
   \[ (40, 10) \]

3. Intersection of $ y \leq 30 $ and $ x + y \leq 50 $:
   \[ x + y = 50 \]
   \[ y = 30 \]
   \[ x + 30 = 50 \implies x = 20 \]
   \[ (20, 30) \]

4. Intersection of $ x + y \leq 50 $ and the axes:
   \[ (0, 0), (50, 0), (0, 50) \]

However, only those vertices that lie within all constraints are relevant:
\[ (0, 0), (40, 10), (20, 30) \]

### Evaluating the Objective Function at the Vertices:
1. At $ (0, 0) $:
   \[ P = 20(0) + 30(0) = 0 \]

2. At $ (40, 10) $:
   \[ P = 20(40) + 30(10) = 800 + 300 = 1100 \]

3. At $ (20, 30) $:
   \[ P = 20(20) + 30(30) = 400 + 900 = 1300 \]

The maximum profit occurs at the vertex $ (20, 30) $.

### Conclusion:
The number of each gadget to be manufactured to maximize the profit is $ 20 $ regular gadgets and $ 30 $ premium gadgets.

So, the ordered pair is:
\[ (20, 30) \]

Would you like any more details or have any questions about this solution?

### Additional Questions:
1. How would the solution change if the profit per regular gadget increased to \$25?
2. What if the total production constraint was reduced to 45 gadgets?
3. How would the solution be affected if a new constraint $ x \geq 10 $ was added?
4. What are the implications if the maximum number of regular gadgets was increased to 50?
5. How would changes in the profit margin of premium gadgets affect the production strategy?
6. Can we find an optimal solution if one of the gadgets could only be produced in even quantities?
7. How does changing the constraints to $ x + 2y \leq 60 $ affect the solution?
8. What would be the optimal production strategy if the objective function changed to $ P = 25x + 20y $?

### Tip:
When solving linear programming problems, always ensure all

Learn how to maximize profit by manufacturing regular and premium gadgets using linear programming techniques. Solve the problem with detailed steps and understand the implications of various constraints and objective functions.

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