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We are given the following problem:

A mining company has two mines. The mines produce two grades of ore: regular-grade and low-grade. The company must produce at least **180 tons** of regular-grade ore and **400 tons** of low-grade ore per week. The first mine produces **2 tons of regular-grade** and **8 tons of low-grade ore per hour**, and can be operated for at most **60 hours** per week. The second mine produces **9 tons of regular-grade** and **5 tons of low-grade ore per hour**, and can be operated for no more than **40 hours** per week.

Let:
- \( x \) be the number of hours the first mine is operated per week.
- \( y \) be the number of hours the second mine is operated per week.

### Constraints:
1. The regular-grade ore requirement:
   \[
   2x + 9y \geq 180
   \]
2. The low-grade ore requirement:
   \[
   8x + 5y \geq 400
   \]
3. Operating time constraints:
   \[
   x \leq 60 \quad \text{(since the first mine cannot operate more than 60 hours)}
   \]
   \[
   y \leq 40 \quad \text{(since the second mine cannot operate more than 40 hours)}
   \]
4. Non-negative constraints:
   \[
   x \geq 0 \quad \text{and} \quad y \geq 0
   \]

These inequalities describe a system of constraints. To visualize the region that satisfies all these constraints, we would graph these inequalities on the coordinate plane with \( x \) and \( y \) as the axes. The feasible region (the shaded area) will represent the combinations of \( x \) and \( y \) that satisfy all the conditions.

Would you like me to graph the solution or help further analyze the constraints?

A mining company owns two mines. These mines produce an ore that can be graded into two classes: regular grade and low grade. The company must produce at least 180 tons of regular-grade and 400 tons of low-grade ore per week. The first mine produces 2 tons of regular-grade and 8 tons of low-grade ore per hour. The first mine can be operated no more than 60 hours a week, and the second mine produces 9 tons of regular-grade and 5 tons of low-grade ore per hour. The second mine can be operated no more than 40 hours a week. Let x be the number of hours the first mine is operated. Let y be the number of hours the second mine is operated. Shade the region corresponding to all values of x and y that satisfy these requirements.

Explore how linear programming is used to determine the operational hours of two mines to meet production requirements of different ore grades. This problem uses systems of linear inequalities to find a feasible region satisfying all constraints. Suitable for college level students studying linear programming or operational research.

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