A linear programming model has been formulated for two products, X and Y. The objective
function is depicted by the formula C = 5X + 6Y, where C = contribution, X = the number of
product X to be produced and Y = the number of product Y to be produced.
Each unit of X uses 3kg of material Z and each unit of Y uses 3 kg of material 2. The standard cost of material Z is $2 per kg.
The shadow price for material Z has been worked out and found to be $2.80 per kg.
If an extra 20 kg of material Z becomes available at $2 per kg, what will the maximum increase in contribution be?

To determine the maximum increase in contribution if an extra 20 kg of material Z becomes available, we can use the shadow price. The shadow price represents the additional contribution that would be gained per additional unit of the resource (in this case, per kilogram of material Z).

### Given Data:
- Objective function: $ C = 5X + 6Y $
- Material Z requirement: 3 kg per unit of product X and 3 kg per unit of product Y
- Standard cost of material Z: $2 per kg
- Shadow price for material Z: $2.80 per kg
- Extra material Z available: 20 kg

### Step-by-Step Calculation:

1. **Shadow Price Meaning:**
   The shadow price of $2.80 per kg means that for every additional kg of material Z available, the contribution (C) increases by $2.80.

2. **Maximum Increase in Contribution:**
   If 20 kg of material Z becomes available, the maximum increase in contribution can be calculated as:
   \[
   \text{Maximum Increase in Contribution} = \text{Shadow Price} \times \text{Extra kg of Material Z}
   \]
   Substituting the values:
   \[
   \text{Maximum Increase in Contribution} = 2.80 \, \text{\$/kg} \times 20 \, \text{kg} = 56 \, \text{\$}
   \]

### Conclusion:
The maximum increase in contribution if an extra 20 kg of material Z becomes available at $2 per kg is **$56**.

Would you like more details on any part of the solution, or have any questions?

Here are some related questions:
1. How does the shadow price influence decision-making in linear programming?
2. What if the shadow price of material Z was less than the standard cost—what would be the impact?
3. How would this scenario change if the extra material Z was available at $3 per kg?
4. Can shadow prices be negative? What would that imply in a linear programming model?
5. How can changes in the availability of other resources affect the overall contribution?

**Tip:** Shadow prices provide insight into the value of scarce resources in a linear programming model, helping to make informed decisions about resource allocation.

Explore how to maximize contribution using shadow prices in linear programming. Understand the impact of additional resources on the objective function C = 5X + 6Y, with a detailed example calculation involving material Z. Suitable for advanced college-level students and professionals.

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