A farmer wants to plant two types of crops, Wheat and Corn, on his land. Each acre of Wheat requires 2 units of fertilizer and 3 units of water, while each acre of Corn requires 4 units of fertilizer and 2 units of water. The farmer has 40 units of fertilizer and 30 units of water available.

The profit per acre for Wheat is $200, and the profit per acre for Corn is $150.

This problem involves maximizing the farmer's profit under certain resource constraints, which makes it a linear programming problem. Let's set up the decision variables, constraints, and objective function.

### Variables:
Let:
- $ x $ = acres of Wheat planted
- $ y $ = acres of Corn planted

### Objective:
The farmer wants to **maximize profit**. The profit from Wheat is $200 per acre, and the profit from Corn is $150 per acre. Thus, the objective function to maximize is:
\[
\text{Maximize } Z = 200x + 150y
\]
where $ Z $ is the total profit.

### Constraints:
1. **Fertilizer constraint**: Each acre of Wheat requires 2 units of fertilizer, and each acre of Corn requires 4 units of fertilizer. The total available fertilizer is 40 units. Therefore, the constraint is:
   \[
   2x + 4y \leq 40
   \]

2. **Water constraint**: Each acre of Wheat requires 3 units of water, and each acre of Corn requires 2 units of water. The total available water is 30 units. Therefore, the constraint is:
   \[
   3x + 2y \leq 30
   \]

3. **Non-negativity constraint**: Since the number of acres cannot be negative, we have:
   \[
   x \geq 0 \quad \text{and} \quad y \geq 0
   \]

### Linear Programming Model:
\[
\text{Maximize } Z = 200x + 150y
\]
subject to:
\[
\begin{aligned}
2x + 4y &\leq 40 \\
3x + 2y &\leq 30 \\
x &\geq 0 \\
y &\geq 0
\end{aligned}
\]

The next step would be to solve this linear programming problem either graphically or using an optimization technique like the Simplex method. Would you like a detailed solution, or should I help solve it graphically? Let me know if you'd prefer a specific method.

---

**Related Questions:**
1. What would happen if the fertilizer or water constraints changed?
2. How does the optimal solution change if the profit per acre for Corn increases to $180?
3. What if the available resources (fertilizer and water) were doubled?
4. Could the farmer plant only Wheat or only Corn, and what would be the profit?
5. How does the problem change if we introduce more crops?

**Tip**: In linear programming, non-negativity constraints are essential to avoid nonsensical solutions (e.g., planting a negative number of acres).

A farmer wants to plant two types of crops, Wheat and Corn, on his land. Each acre of Wheat requires 2 units of fertilizer and 3 units of water, while each acre of Corn requires 4 units of fertilizer and 2 units of water. The farmer has 40 units of fertilizer and 30 units of water available. The profit per acre for Wheat is $200, and the profit per acre for Corn is $150.

Solve a linear programming problem to maximize profit by planting Wheat and Corn. This problem involves optimizing resources (fertilizer and water) with constraints. Learn to set up the objective function and constraints and understand how to solve the problem using linear programming techniques. Suitable for students in Grades 10-12.

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