A farmer can buy two types of plant​ food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric​ acid, 30 pounds of​ nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric​ acid, 30 pounds of​ nitrogen, and 10 pounds of potash. The minimum monthly requirements are 460 pounds of phosphoric​ acid, 990 pounds of​ nitrogen, and 210 pounds of potash. If mix A costs ​$25 per cubic yard and mix B costs ​$35 per cubic​ yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum​ cost? What is this​ cost?

This problem can be formulated as a linear programming problem where we aim to minimize the cost while meeting the nutrient requirements.

### Variables:
Let:
- $ x $ = the number of cubic yards of mix A
- $ y $ = the number of cubic yards of mix B

### Objective function:
The total cost is given by:
\[
\text{Minimize } C = 25x + 35y
\]
Where $ x $ cubic yards of mix A costs $25 per cubic yard, and $ y $ cubic yards of mix B costs $35 per cubic yard.

### Constraints:
We need to ensure the farmer meets the minimum monthly nutrient requirements for phosphoric acid, nitrogen, and potash. These can be translated into inequalities based on the nutrient content of each mix:

1. **Phosphoric acid constraint**: 
   \[
   20x + 10y \geq 460
   \]
   (20 pounds of phosphoric acid from mix A per cubic yard and 10 pounds from mix B).

2. **Nitrogen constraint**:
   \[
   30x + 30y \geq 990
   \]
   (30 pounds of nitrogen from both mix A and mix B per cubic yard).

3. **Potash constraint**:
   \[
   5x + 10y \geq 210
   \]
   (5 pounds of potash from mix A and 10 pounds from mix B per cubic yard).

Additionally, both $ x $ and $ y $ must be non-negative:
\[
x \geq 0, \quad y \geq 0
\]

### Solution:
To solve this linear programming problem, we can use graphical methods or simplex methods to find the optimal solution. I'll proceed by solving it step by step using linear programming techniques.

Let me calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

However, you can try solving this problem using a linear programming tool like Excel Solver, or I can guide you through the steps manually if you prefer.

Would you like a detailed walkthrough on how to solve it graphically, or using another approach?

Here are 5 related questions:
1. How would you set up a linear programming problem in Excel?
2. Can you explain what the simplex method is used for in optimization?
3. What are the differences between graphical and algebraic methods in linear programming?
4. What are other common applications of linear programming in real-life scenarios?
5. What happens when a linear programming problem has no feasible solution?

**Tip:** In linear programming, always check if the constraints lead to a feasible region before solving for the optimal solution!

A farmer can buy two types of plant food, mix A and mix B. Each cubic yard of mix A contains 20 pounds of phosphoric acid, 30 pounds of nitrogen, and 5 pounds of potash. Each cubic yard of mix B contains 10 pounds of phosphoric acid, 30 pounds of nitrogen, and 10 pounds of potash. The minimum monthly requirements are 460 pounds of phosphoric acid, 990 pounds of nitrogen, and 210 pounds of potash. If mix A costs $25 per cubic yard and mix B costs $35 per cubic yard, how many cubic yards of each mix should the farmer blend to meet the minimum monthly requirements at a minimum cost? What is this cost?

Learn how to solve a linear programming problem to minimize costs for a farmer purchasing plant food mixes A and B while meeting specific nutrient requirements. This example demonstrates key concepts in optimization, using inequalities and the simplex method, suitable for college-level students.

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