Kellogg is a new cereal formed of a mixture of bran and rice, that contains at least 88 grams of protein and at least 36 milligrams of iron. Knowing that bran contains 80 grams of protein and 40 milligrams of iron per kilogram, and that rice contains 100 grams of protein and 30 milligrams of iron per kilogram, find the minimum cost of producting this new cereal if bran costs Rs.5 per kilogram and rice costs Rs.4 per kilogram.

Explanation

To solve the problem of finding the minimum cost of producing the new cereal made from bran and rice while satisfying the protein and iron requirements, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = kilograms of bran
- \( y \) = kilograms of rice

### Step 2: Formulate the Cost Function
The cost of producing the cereal can be expressed as:
\[ Z = 5x + 4y \]
where \( 5 \) is the cost per kilogram of bran and \( 4 \) is the cost per kilogram of rice. We need to minimize \( Z \).

### Step 3: Set Up Constraints
We have two constraints based on the protein and iron requirements:

1. **Protein Constraint**:
- Bran provides 80 grams of protein per kg, and rice provides 100 grams of protein per kg. The total protein must be at least 88 grams.
\[ 80x + 100y \geq 88 \quad \text{(in grams)} \]
Converting to kilograms:
\[ 0.08x + 0.1y \geq 0.088 \quad \text{(in kg)} \]
Multiplying through by 1000:
\[ 80x + 100y \geq 88 \quad \text{(in grams)} \]

2. **Iron Constraint**:
- Bran provides 40 mg of iron per kg, and rice provides 30 mg of iron per kg. The total iron must be at least 36 mg.
\[ 40x + 30y \geq 36 \]

### Step 4: Rewrite Constraints
Rearranging the constraints gives us:
1. \( 80x + 100y \geq 88 \) (Protein)
2. \( 40x + 30y \geq 36 \) (Iron)

### Step 5: Convert to Standard Form
To make it easier to graph, we can rewrite the inequalities:
1. \( 80x + 100y \geq 88 \) can be rewritten as:
\[ 4x + 5y \geq 4.4 \quad \text{(dividing by 20)} \]
2. \( 40x + 30y \geq 36 \) can be rewritten as:
\[ 4x + 3y \geq 3.6 \quad \text{(dividing by 10)} \]

### Step 6: Graph the Constraints
To graph the constraints, we can find the intercepts:
1. For \( 4x + 5y = 4.4 \):
- If \( x = 0 \), \( y = \frac{4.4}{5} = 0.88 \)
- If \( y = 0 \), \( x = \frac{4.4}{4} = 1.1 \)

2. For \( 4x + 3y = 3.6 \):
- If \( x = 0 \), \( y = \frac{3.6}{3} = 1.2 \)
- If \( y = 0 \), \( x = \frac{3.6}{4} = 0.9 \)

### Step 7: Identify Feasible Region
The feasible region is where the constraints overlap, and it must be in the first quadrant (since \( x \) and \( y \) cannot be negative).

### Step 8: Find Corner Points
The corner points of the feasible region can be found by solving the equations of the lines:
1. Solve \( 4x + 5y = 4.4 \) and \( 4x + 3y = 3.6 \):
- Subtract the second from the first:
\[ (4x + 5y) - (4x + 3y) = 4.4 - 3.6 \implies 2y = 0.8 \implies y = 0.4 \]
Substitute \( y = 0.4 \) into \( 4x + 3(0.4) = 3.6 \):
\[ 4x + 1.2 = 3.6 \implies 4x = 2.4 \implies x = 0.6 \]
So one corner point is \( (0.6, 0.4) \).

2. Find other corner points by checking intersections with axes.

### Step 9: Evaluate Cost Function at Corner Points
Evaluate \( Z = 5x + 4y \) at each corner point:
- For \( (0.6, 0.4) \):
\[ Z = 5(0.6) + 4(0.4) = 3 + 1.6 = 4.6 \]

### Step 10: Determine Minimum Cost
The minimum cost occurs at \( (0.6, 0.4) \) with a cost of Rs. 4.6.

### Final Answer
The minimum cost of producing the new cereal is Rs. 4.6.

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