A pork farmer plans to mix two types of food to make feed for the pigs he is raising. A bag of Formula X costs $25 and contains 32 units of protein, 29 units of fat, and 25 units of fiber. A bag of Formula Y costs $20 and contains 30 units of protein, 21 units of fat, and 24 units of fiber.

How many bags of Formula X and Y should the consumed by the pigs each day in order to meet the minimum daily requirements of 220 units of protein, 170 units of fat, and 180 units of fiber at a minimum cost?

(a)

Identify and clearly define the variables needed to solve the linear programming problem. C=?

To formulate the linear programming problem, we start by defining the variables. Since the goal is to minimize the cost of the feed, we need to determine how many bags of Formula X and Formula Y the farmer should use to meet the daily requirements for protein, fat, and fiber.

### Variables:
Let:
- $ x $ = the number of bags of Formula X used per day.
- $ y $ = the number of bags of Formula Y used per day.

### Objective:
The objective is to minimize the total cost of the feed, which can be expressed as:
\[ C = 25x + 20y \]
Where:
- $ 25x $ is the cost for $ x $ bags of Formula X.
- $ 20y $ is the cost for $ y $ bags of Formula Y.

### Constraints:
The farmer needs to ensure the pigs meet their minimum daily nutritional requirements. These requirements translate into the following constraints:

1. **Protein Requirement**:
   \[
   32x + 30y \geq 220
   \]
   The combined protein content from $ x $ bags of Formula X and $ y $ bags of Formula Y should be at least 220 units.

2. **Fat Requirement**:
   \[
   29x + 21y \geq 170
   \]
   The combined fat content from both formulas must be at least 170 units.

3. **Fiber Requirement**:
   \[
   25x + 24y \geq 180
   \]
   The combined fiber content from both formulas must be at least 180 units.

4. **Non-Negativity Constraint**:
   \[
   x \geq 0, \, y \geq 0
   \]
   The number of bags cannot be negative.

### Summary:
The variables needed to solve the linear programming problem are $ x $ (the number of bags of Formula X) and $ y $ (the number of bags of Formula Y).

**Objective Function**: Minimize $ C = 25x + 20y $.

**Constraints**:
1. $ 32x + 30y \geq 220 $ (Protein)
2. $ 29x + 21y \geq 170 $ (Fat)
3. $ 25x + 24y \geq 180 $ (Fiber)
4. $ x \geq 0, \, y \geq 0 $ (Non-negativity)

---

Would you like to see the solution to this problem using a specific method? Any questions?

Here are some related questions you could explore:
1. How can the graphical method be used to solve this linear programming problem?
2. What is the importance of the non-negativity constraint in this context?
3. How do the constraints define the feasible region in this problem?
4. Can the Simplex method be applied here, and how does it work?
5. How would the solution change if the cost of Formula X or Y were to increase?

**Tip**: Always check if all constraints are satisfied after obtaining a solution, especially in optimization problems!

A pork farmer plans to mix two types of food to make feed for the pigs he is raising. A bag of Formula X costs $25 and contains 32 units of protein, 29 units of fat, and 25 units of fiber. A bag of Formula Y costs $20 and contains 30 units of protein, 21 units of fat, and 24 units of fiber. How many bags of Formula X and Y should the farmer use each day to meet the minimum daily requirements of 220 units of protein, 170 units of fat, and 180 units of fiber at a minimum cost?

Learn how to solve a linear programming problem to minimize feed costs for pigs while meeting daily protein, fat, and fiber requirements. This step-by-step guide explains how to use the cost function and constraints to find the optimal number of bags of Formula X and Y. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation