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The question in the image involves a linear programming problem related to maximizing revenue under weight and space constraints for a plane transporting food and water to a resort island. Here's a breakdown of the problem:

### Problem Summary:
- The plane can carry a maximum of **27,000 pounds** and no more than **550 cubic feet** of cargo.
- **Each container of water**:
  - Weighs **60 pounds** and takes up **1 cubic foot** of cargo.
- **Each container of food**:
  - Weighs **150 pounds** and takes up **5 cubic feet** of cargo.
- Hotels will buy:
  - Food for **$17 per pound**.
  - Water for **$4 per pound**.

Let:
- $ x $ be the number of **food containers**.
- $ y $ be the number of **water containers**.

We need to determine:
- The optimal numbers of containers ($x$ for food and $y$ for water) to maximize the revenue, while respecting the weight and space limitations of the plane.

### Constraints:
1. **Weight Constraint**: 
   \[
   150x + 60y \leq 27,000
   \]
2. **Space Constraint**:
   \[
   5x + 1y \leq 550
   \]

### Objective Function (Revenue):
The total revenue $ R $ generated is the sum of the revenue from food and water:
\[
R = 17(150x) + 4(60y) = 2550x + 240y
\]

### Solution Method:
To solve this, you would typically:
1. Graph the constraints on an $xy$-plane.
2. Identify the feasible region.
3. Calculate the value of the objective function at the vertices of the feasible region.
4. The vertex with the highest value is the solution for maximizing revenue.

Would you like me to walk through a detailed solution or explain a specific part of the process?

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Here are 5 follow-up questions related to this problem:
1. How do you graph the weight and space constraints on the $xy$-plane?
2. What is the significance of the feasible region in linear programming?
3. How do you calculate the vertices of the feasible region?
4. What happens if the maximum revenue occurs at more than one point?
5. How can the Simplex Method be used to solve this linear programming problem?

**Tip**: Always check whether your constraints form a bounded feasible region, as an unbounded region may not lead to a maximum solution in some cases.

A plane carrying food and water to a resort island can carry a maximum of 27000 pounds and is limited to space for no more than 550 cubic feet. Each container of water weighs 60 pounds and takes up 1 cubic foot of cargo space. Each container of food weighs 150 pounds and takes up 5 cubic feet. Hotels on the island will buy food for 17 dollars a pound and water for 4 dollars a pound. What is the optimum number of containers of each item that will maximize the revenue generated by the plane? What is the maximum revenue? Let x represent the number of food containers and y represent the number of water containers.

Learn how to solve a linear programming problem to maximize revenue for a cargo plane carrying food and water. This problem involves optimizing under weight and space constraints, with detailed step-by-step solutions using systems of inequalities. Suitable for advanced high school or college-level students.

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