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To set up this linear programming (LP) problem, let's analyze the objective function and constraints based on the given dataTo set up this linear programming (LP) problem, let's analyze the objective function and constraints based on the given 

### 1. Define Variables
- Let $ x $ be the number of days Mine 1 operates.
- Let $ y $ be the number of days Mine 2 operates.
- Let $ z $ be the number of days Mine 3 operates.

### 2. Objective Function
The goal is to minimize the total cost of operating the mines, where:
- Mine 1 costs $7,300 per day,
- Mine 2 costs $6,900 per day,
- Mine 3 costs $6,700 per day.

Thus, the objective function to minimize the total cost is:
\[
\text{Minimize Cost} = 7300x + 6900y + 6700z
\]

### 3. Constraints
The company must meet minimum ore production requirements for iron, lead, and nickel:

#### Iron Constraint
From the table:
- Mine 1 produces 25 tons/day,
- Mine 2 produces 35 tons/day,
- Mine 3 produces 25 tons/day.

The total production for iron needs to be at least 1550 tons:
\[
25x + 35y + 25z \geq 1550
\]

#### Lead Constraint
- Mine 1 produces 15 tons/day,
- Mine 2 produces 0 tons/day,
- Mine 3 produces 20 tons/day.

The total production for lead needs to be at least 760 tons:
\[
15x + 0y + 20z \geq 760
\]

#### Nickel Constraint
- Mine 1 produces 10 tons/day,
- Mine 2 produces 25 tons/day,
- Mine 3 produces 0 tons/day.

The total production for nickel needs to be at least 510 tons:
\[
10x + 25y + 0z \geq 510
\]

### 4. Solution
To find the number of days ($ x $, $ y $, $ z $) that minimizes cost, we can solve this LP problem using methods like the Simplex algorithm or software tools. 

### Summary
1. **Objective Function:** $\text{Minimize Cost} = 7300x + 6900y + 6700z$
2. **Constraints:**
   - $25x + 35y + 25z \geq 1550$
   - $15x + 0y + 20z \geq 760$
   - $10x + 25y + 0z \geq 510$

If you'd like, I can walk you through the process of solving it, or I can show you how to use software to solve it.

Would you like more details, or do you have any questions?

---

Here are five additional questions for practice:
1. What would the objective function be if the costs of each mine changed?
2. How would the constraints change if the required tons for each ore type were higher?
3. How would the solution change if Mine 2 could produce lead?
4. What if the requirement for nickel was reduced to 400 tons?
5. How would you solve this LP problem if there were a fourth mine?

**Tip:** In linear programming, always clearly define your variables and constraints before setting up the objective function. This ensures accuracy and simplifies solving.

MJ Mining Company owns three mines and has three different types of ores that is mined. For each ore, the number of tons per day available from each mine and the minimum number of tons required to fill orders are given in the following table. Let x, y, and z represent the number of days to run Mines 1, 2, and 3 respectively. Type the objective function and the left-hand side of each constraint in the following answer boxes.

Solve a linear programming problem to minimize costs for MJ Mining Company. Given the daily operation costs and ore production capacities of three mines, determine the optimal number of days each mine should operate to meet ore demands for iron, lead, and nickel while minimizing expenses.

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