Question 1176614
Let's set up and solve this linear programming problem.

**1. Define Variables**

* Let 't' be the number of teachers.
* Let 'a' be the number of aides.

**2. Formulate the Objective Function (Cost)**

The objective is to minimize the cost. The cost function (C) is:

* C = 30000t + 20000a

**3. Formulate the Constraints**

* **Total Faculty Constraint:** t + a ≤ 50
* **Minimum Faculty Constraint:** t + a ≥ 20
* **Minimum Teachers Constraint:** t ≥ 12
* **Teacher-to-Aide Ratio Constraint:** t ≥ (2/3)a
* **Non-negativity Constraints:** t ≥ 0, a ≥ 0

**4. Simplify the Teacher-to-Aide Ratio Constraint**

* 3t ≥ 2a
* a ≤ (3/2)t

**5. Find the Corner Points**

We need to graph the constraints and find the intersection points.

* **Constraint 1 (t + a ≤ 50):** a ≤ 50 - t
* **Constraint 2 (t + a ≥ 20):** a ≥ 20 - t
* **Constraint 3 (t ≥ 12):** Vertical line at t = 12
* **Constraint 4 (a ≤ (3/2)t):** a ≤ 1.5t

Let's find the intersections:

* **Intersection of t = 12 and a = 20 - t:** a = 20 - 12 = 8. Point (12, 8)
* **Intersection of t = 12 and a = 1.5t:** a = 1.5 * 12 = 18. Point (12, 18)
* **Intersection of t = 12 and a = 50 - t:** a = 50 - 12 = 38. Point (12, 38)
* **Intersection of a = 1.5t and a = 50 - t:** 1.5t = 50 - t, 2.5t = 50, t = 20, a = 1.5 * 20 = 30. Point (20, 30)
* **Intersection of a = 20 - t and a = 1.5t:** 20 - t = 1.5t, 20 = 2.5t, t = 8. This violates t>=12. Therefore, this intersection is not in the feasible region.
* **Intersection of a = 20 - t and a = 50 - t:** Not possible, as they are parallel.

The corner points of the feasible region are:

* (12, 8)
* (12, 18)
* (12, 38)
* (20, 30)

**6. Evaluate the Objective Function at Each Corner Point**

* **(12, 8):** C = 30000(12) + 20000(8) = 360000 + 160000 = 520000
* **(12, 18):** C = 30000(12) + 20000(18) = 360000 + 360000 = 720000
* **(12, 38):** C = 30000(12) + 20000(38) = 360000 + 760000 = 1120000
* **(20, 30):** C = 30000(20) + 20000(30) = 600000 + 600000 = 1200000

**7. Determine the Minimum Cost**

The minimum cost is $520,000, which occurs at the point (12, 8).

**Answers**

* **Teachers:** 12
* **Aides:** 8
* **Objective Function (Cost):** C = 30000t + 20000a