Formulate but do not solve the following exercise as a linear programming problem.

A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is $50/acre, whereas the cost of cultivating Crop B is $70/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of Crop A requires 20 labor-hours, and each acre of Crop B requires 25 labor-hours. The farmer has a maximum of 2900 labor-hours available. If she expects to make a profit of $160/acre on Crop A and $250/acre on Crop B, how many acres of each crop, x and y, respectively, should she plant to maximize her profit P in dollars? Skip the steps and show the answers

The linear programming formulation for this problem is as follows:

### Decision Variables:
- Let $ x $ represent the number of acres of Crop A to be planted.
- Let $ y $ represent the number of acres of Crop B to be planted.

### Objective Function:
- Maximize profit $ P $, where:
  \[
  P = 160x + 250y
  \]
  (since the profit per acre is $160 for Crop A and $250 for Crop B).

### Constraints:
1. **Budget constraint (Cultivation costs)**: 
   \[
   50x + 70y \leq 7400
   \]
   (since the farmer has a maximum of $7400 for cultivation costs, with Crop A costing $50/acre and Crop B costing $70/acre).

2. **Labor constraint**:
   \[
   20x + 25y \leq 2900
   \]
   (since the farmer has a maximum of 2900 labor-hours available, with Crop A requiring 20 labor-hours/acre and Crop B requiring 25 labor-hours/acre).

3. **Non-negativity constraints**:
   \[
   x \geq 0, \quad y \geq 0
   \]
   (since the number of acres planted cannot be negative).

### Summary:
- **Maximize**: $ P = 160x + 250y $
- **Subject to**:
  \[
  50x + 70y \leq 7400
  \]
  \[
  20x + 25y \leq 2900
  \]
  \[
  x \geq 0, \quad y \geq 0
  \]

This is the linear programming formulation of the problem. 

Would you like any further details or explanations?

### Expanding on this:
1. How would changes in profit per acre impact the optimal solution?
2. What would happen if the labor-hours constraint increased?
3. Could the model be adjusted to account for additional crop types?
4. How would the problem change if there were additional budget limitations?
5. What methods can be used to solve this linear programming problem efficiently?

**Tip**: In linear programming, sensitivity analysis can help understand how changes in constraints or the objective function affect the optimal solution.

Formulate but do not solve the following exercise as a linear programming problem. A farmer plans to plant two crops, A and B. The cost of cultivating Crop A is $50/acre, whereas the cost of cultivating Crop B is $70/acre. The farmer has a maximum of $7400 available for land cultivation. Each acre of Crop A requires 20 labor-hours, and each acre of Crop B requires 25 labor-hours. The farmer has a maximum of 2900 labor-hours available. If she expects to make a profit of $160/acre on Crop A and $250/acre on Crop B, how many acres of each crop, x and y, respectively, should she plant to maximize her profit P in dollars? Skip the steps and show the answers.

Learn how to formulate a linear programming problem where a farmer must decide how many acres of two crops, A and B, to plant to maximize profits, considering budget and labor constraints. This problem explores optimization techniques and is suitable for advanced high school or undergraduate students.

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