Question 1176370
To answer this question, we need the original Electrocomp linear programming problem. I'll assume it's a typical production problem with:

* **Variables:**
    * x = number of air conditioners
    * y = number of fans
* **Objective Function (Maximize Profit):**
    * Let's assume it's something like P = ax + by (where a and b are profit per air conditioner and fan, respectively).
* **Constraints (Example):**
    * Resource 1: c1x + d1y ≤ e1
    * Resource 2: c2x + d2y ≤ e2
    * x ≥ 0, y ≥ 0

**Important:** You must replace these general constraints and objective function with the actual Electrocomp problem constraints and objective function.

**Now, let's address the new constraints:**

**(a) At least 20 air conditioners (x ≥ 20) and no more than 80 fans (y ≤ 80)**

1.  **Add the new constraints:**
    * x ≥ 20
    * y ≤ 80

2.  **Graph the feasible region:**
    * Graph all the original constraints and the two new ones.
    * Identify the corner points of the new feasible region.

3.  **Evaluate the objective function:**
    * Calculate the profit (P) at each corner point.
    * Determine the corner point that yields the maximum profit.

4.  **Calculate slack:**
    * For each constraint, plug the optimal (x, y) values.
    * If the constraint is "≤", slack is the difference between the right-hand side and the left-hand side.
    * If the constraint is "≥", slack is the difference between the left-hand side and the right-hand side.

**(b) At least 30 air conditioners (x ≥ 30) and no more than 50 fans (y ≤ 50)**

1.  **Add the new constraints:**
    * x ≥ 30
    * y ≤ 50

2.  **Graph the feasible region:**
    * Graph all the original constraints and the two new ones.
    * Identify the corner points of the new feasible region.

3.  **Evaluate the objective function:**
    * Calculate the profit (P) at each corner point.
    * Determine the corner point that yields the maximum profit.

4.  **Calculate slack:**
    * For each constraint, plug the optimal (x, y) values.
    * If the constraint is "≤", slack is the difference between the right-hand side and the left-hand side.
    * If the constraint is "≥", slack is the difference between the left-hand side and the right-hand side.

**Example (Illustrative - You must replace with actual constraints):**

Let's assume the example Electrocomp problem is:

* Maximize P = 20x + 15y
* Constraints:
    * x + y ≤ 100 (Resource 1)
    * 2x + y ≤ 150 (Resource 2)
    * x ≥ 0, y ≥ 0

**(a) x ≥ 20, y ≤ 80**

1.  **Feasible region:** The feasible region is bounded by the original constraints and the new ones.

2.  **Corner points (after graphing):** (20, 80), (20, 60), (35, 65), (75, 0)

3.  **P values:**
    * (20, 80): P = 20(20) + 15(80) = 1600
    * (20, 60): P = 20(20) + 15(60) = 1300
    * (35, 65): P = 20(35) + 15(65) = 1675
    * (75, 0): P = 20(75) + 15(0) = 1500

4.  **Optimal solution:** (35, 65), P = 1675

5.  **Slack:**
    * x + y ≤ 100: 35 + 65 = 100 (slack = 0)
    * 2x + y ≤ 150: 2(35) + 65 = 135 (slack = 15)
    * x ≥ 20: 35 - 20 = 15 (slack = 15)
    * y ≤ 80: 80 - 65 = 15 (slack = 15)

**(b) x ≥ 30, y ≤ 50**

Repeat the process with the new constraints.