Minimize (16-21): OBJECTIVE FUNCTION CONSTRAINTS
`Z = 200x + 500y`
`x + 2y ge 10, 3x + 4y le 24, x ge 0, y ge 0`.

Answer

To solve the given linear programming problem, we will follow these steps:

### Step 1: Define the Objective Function and Constraints
We need to minimize the objective function:
\[ Z = 200x + 500y \]

Subject to the constraints:
1. \( x + 2y \geq 10 \)
2. \( 3x + 4y \leq 24 \)
3. \( x \geq 0 \)
4. \( y \geq 0 \)

### Step 2: Convert Inequalities to Equations
To find the boundary lines of the constraints, we convert the inequalities into equations:

1. For \( x + 2y = 10 \)
2. For \( 3x + 4y = 24 \)

### Step 3: Find the Intercepts of the Constraints
To graph these equations, we find the x-intercepts and y-intercepts.

**For \( x + 2y = 10 \):**
- When \( x = 0 \): \( 2y = 10 \) → \( y = 5 \) (y-intercept)
- When \( y = 0 \): \( x = 10 \) (x-intercept)

**For \( 3x + 4y = 24 \):**
- When \( x = 0 \): \( 4y = 24 \) → \( y = 6 \) (y-intercept)
- When \( y = 0 \): \( 3x = 24 \) → \( x = 8 \) (x-intercept)

### Step 4: Plot the Constraints
Now we can plot these intercepts on a graph:
- The line \( x + 2y = 10 \) intersects the axes at (10, 0) and (0, 5).
- The line \( 3x + 4y = 24 \) intersects the axes at (8, 0) and (0, 6).

### Step 5: Identify the Feasible Region
The feasible region is where the constraints overlap. We will check the direction of the inequalities:
- For \( x + 2y \geq 10 \): The region is above the line.
- For \( 3x + 4y \leq 24 \): The region is below the line.
- Both \( x \geq 0 \) and \( y \geq 0 \) restrict us to the first quadrant.

### Step 6: Find the Corner Points of the Feasible Region
The corner points of the feasible region can be found as follows:
1. Intersection of \( x + 2y = 10 \) and \( 3x + 4y = 24 \).

To find the intersection, we can solve the equations simultaneously:
- From \( x + 2y = 10 \), we can express \( x = 10 - 2y \).
- Substitute into \( 3(10 - 2y) + 4y = 24 \):
\[ 30 - 6y + 4y = 24 \]
\[ -2y = -6 \]
\[ y = 3 \]
- Substitute \( y = 3 \) back into \( x + 2(3) = 10 \):
\[ x + 6 = 10 \]
\[ x = 4 \]

Thus, the intersection point is \( (4, 3) \).

### Step 7: Evaluate the Objective Function at Each Corner Point
Now we evaluate \( Z = 200x + 500y \) at the corner points:
1. At \( (0, 6) \):
\[ Z = 200(0) + 500(6) = 3000 \]
2. At \( (0, 5) \):
\[ Z = 200(0) + 500(5) = 2500 \]
3. At \( (4, 3) \):
\[ Z = 200(4) + 500(3) = 800 + 1500 = 2300 \]

### Step 8: Determine the Minimum Value
The minimum value of \( Z \) occurs at the point \( (4, 3) \):
\[ \text{Minimum value of } Z = 2300 \]

### Final Answer
The minimum value of the objective function \( Z \) is \( 2300 \) at the point \( (4, 3) \).

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