Refers to question 12. i.e Z represents cost , if `Z=400x+200y, x+2y ≥ 30`, `x-y=0` and `5x+2y ≥ 30`, What will be the minimum cost?

Explanation

To solve the given problem step by step, we will follow these guidelines:

### Given:
- Objective function: \( Z = 400x + 200y \)
- Constraints:
1. \( x + 2y \geq 30 \)
2. \( x - y = 0 \) (which implies \( x = y \))
3. \( 5x + 2y \geq 30 \)

### Step 1: Rewrite the constraints
From the equations, we can rewrite the constraints:
1. \( x + 2y \geq 30 \) (Constraint 1)
2. \( x = y \) (Constraint 2)
3. \( 5x + 2y \geq 30 \) (Constraint 3)

### Step 2: Substitute \( y \) in terms of \( x \)
From Constraint 2, we know \( y = x \). We will substitute \( y \) in the other constraints.

1. Substitute \( y = x \) into Constraint 1:
\[ x + 2x \geq 30 \implies 3x \geq 30 \implies x \geq 10 \]

2. Substitute \( y = x \) into Constraint 3:
\[ 5x + 2x \geq 30 \implies 7x \geq 30 \implies x \geq \frac{30}{7} \approx 4.29 \]

### Step 3: Identify feasible region
From the constraints, we have two inequalities:
- \( x \geq 10 \)
- \( x \geq \frac{30}{7} \)

Since \( 10 \) is greater than \( \frac{30}{7} \), the feasible region is defined by \( x \geq 10 \).

### Step 4: Determine the coordinates of the vertices
Now we need to find the intersection points of the constraints to determine the vertices of the feasible region.

1. From \( x = y \) and \( x + 2y = 30 \):
\[ x + 2x = 30 \implies 3x = 30 \implies x = 10, y = 10 \quad \text{(Point A: (10, 10))} \]

2. From \( x = y \) and \( 5x + 2y = 30 \):
\[ 5x + 2x = 30 \implies 7x = 30 \implies x = \frac{30}{7}, y = \frac{30}{7} \quad \text{(Point B: }\left(\frac{30}{7}, \frac{30}{7}\right)\text{)} \]

### Step 5: Evaluate the objective function at the vertices
Now we will evaluate \( Z = 400x + 200y \) at the vertices:

1. At Point A (10, 10):
\[ Z = 400(10) + 200(10) = 4000 + 2000 = 6000 \]

2. At Point B \(\left(\frac{30}{7}, \frac{30}{7}\right)\):
\[ Z = 400\left(\frac{30}{7}\right) + 200\left(\frac{30}{7}\right) = \frac{12000}{7} + \frac{6000}{7} = \frac{18000}{7} \approx 2571.43 \]

### Step 6: Determine the minimum cost
The minimum cost occurs at Point B, which gives:
\[ \text{Minimum cost } Z \approx 2571.43 \]

### Final Answer:
The minimum cost is approximately \( \boxed{2571.43} \).

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