A school board is investigating various ways of composing the staff for a proposed new elementary school.  They can hire teachers and assistants.  The amount of money the school district will spend on salaries each year depends on how many teachers and on how many assistants are hired.

The board finds that the annual teacher salary is $15,000, and the average assistant salary is $10,000.

The building can accommodate no more than 50 staff members in total. A minimum of 20 staff members is needed to run the school.

The school cannot be run entirely by assistants, so there must be at least 11 teachers. For the proper teacher-to-assistant ratio, the number of teachers must be at least half the number of assistants. It is impossible to hire a negative number of assistants or teachers.

Formulate the linear programming model and then answer questions below.

a.  Determine how many teachers will need to hire to minimize salary cost? *(answer in whole number)*

To formulate the linear programming model for this problem, we will define the variables, constraints, and the objective function.

### Step 1: Define the Variables
Let:
- $ x $ represent the number of teachers hired.
- $ y $ represent the number of assistants hired.

### Step 2: Constraints
The problem gives several constraints:
1. **Total number of staff**: The total number of teachers and assistants combined cannot exceed 50 staff members, and there must be at least 20 staff members. 
   \[
   20 \leq x + y \leq 50
   \]
   
2. **Minimum number of teachers**: There must be at least 11 teachers.
   \[
   x \geq 11
   \]

3. **Teacher-to-assistant ratio**: The number of teachers must be at least half the number of assistants. 
   \[
   x \geq \frac{y}{2}
   \]
   This can be rewritten as:
   \[
   2x \geq y
   \]

4. **Non-negative staff**: It is impossible to hire a negative number of teachers or assistants.
   \[
   x \geq 0, \, y \geq 0
   \]

### Step 3: Objective Function
The objective is to **minimize the salary cost**. The salary for each teacher is $15,000 and for each assistant is $10,000. So, the total cost function $ C $ is:
\[
C = 15,000x + 10,000y
\]
We aim to minimize this cost function.

### Linear Programming Model:
\[
\text{Minimize } C = 15,000x + 10,000y
\]
Subject to:
\[
20 \leq x + y \leq 50
\]
\[
x \geq 11
\]
\[
2x \geq y
\]
\[
x \geq 0, \, y \geq 0
\]

### Step 4: Solve the Linear Programming Model
We now solve the linear program to find the values of $ x $ (number of teachers) and $ y $ (number of assistants) that minimize the salary cost.

To do this, we examine the feasible region defined by the constraints and check corner points. The corner points (extreme points) of the feasible region are the ones where constraints intersect, and the minimum salary cost will occur at one of these points.

Let's calculate the corner points:
1. From $ 2x \geq y $ and $ x + y = 50 $, solving these will give:
   $ 2x \geq y $ -> $ y = 2x $

To minimize salary

A school board is investigating various ways of composing the staff for a proposed new elementary school. They can hire teachers and assistants. The amount of money the school district will spend on salaries each year depends on how many teachers and on how many assistants are hired. The board finds that the annual teacher salary is $15,000, and the average assistant salary is $10,000. The building can accommodate no more than 50 staff members in total. A minimum of 20 staff members is needed to run the school. The school cannot be run entirely by assistants, so there must be at least 11 teachers. For the proper teacher-to-assistant ratio, the number of teachers must be at least half the number of assistants. It is impossible to hire a negative number of assistants or teachers. Formulate the linear programming model and then answer questions below. a. Determine how many teachers will need to hire to minimize salary cost? (answer in whole number)

Discover how to minimize salary costs while staffing a new elementary school. This problem involves formulating and solving a linear programming model, using constraints such as staff limits, teacher-to-assistant ratios, and salary costs. Suitable for Grades 10-12.

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