The county officials in Chang Qing County, China used linear programming to aid the farmers in their choices of crops and other forms of agricultural production. Suppose and American farmer has 480 acres on which to grow corn and soybeans. He is planting at least 40 acres of corn and 20 of soybeans. Based on his calculations, he can earn $150 per acre of corn and $250 acre of soybeans, how many acres of each should he plant of earn the greatest profit? What is the farmer's maximum profit?

We can solve this problem using **linear programming** by defining the variables, objective function, and constraints.

### Step 1: Define the variables
Let:
- $ x $ = acres of **corn** planted
- $ y $ = acres of **soybeans** planted

### Step 2: Write the objective function
The farmer wants to **maximize profit**. The profit is calculated by:
- \$150 per acre of corn
- \$250 per acre of soybeans

Thus, the objective function (profit $ P $) is:
\[
P = 150x + 250y
\]
### Step 3: Write the constraints
- Total available land: $ x + y \leq 480 $ (since the total land available is 480 acres)
- Minimum acreage of corn: $ x \geq 40 $ (since at least 40 acres of corn must be planted)
- Minimum acreage of soybeans: $ y \geq 20 $ (since at least 20 acres of soybeans must be planted)
- Non-negative acreage: $ x \geq 0 $, $ y \geq 0 $

Thus, the system of inequalities (constraints) is:
\[
\begin{aligned}
1. & \quad x + y \leq 480 \\
2. & \quad x \geq 40 \\
3. & \quad y \geq 20 \\
4. & \quad x \geq 0 \\
5. & \quad y \geq 0
\end{aligned}
\]

### Step 4: Solve graphically or using simplex method
We will solve this system to find the maximum profit by finding the corner points (vertices) of the feasible region.

Let me calculate the values of $x$ and $y$ that maximize the profit.The farmer should plant:
- **40 acres of corn** 
- **440 acres of soybeans**

This will give the farmer a **maximum profit of $116,000**.

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How would the solution change if the price per acre of corn increased?
2. What if the farmer had only 400 acres of land available instead of 480?
3. How would the profit be affected if the farmer had to plant at least 100 acres of corn?
4. Can this problem be solved using a different optimization technique, like the graphical method?
5. How can sensitivity analysis help the farmer in case the prices fluctuate?

**Tip:** When solving linear programming problems, it's useful to understand both the feasible region (where constraints overlap) and the vertices of this region, as these points are where optimal solutions often lie.

The county officials in Chang Qing County, China used linear programming to aid the farmers in their choices of crops and other forms of agricultural production. Suppose an American farmer has 480 acres on which to grow corn and soybeans. He is planting at least 40 acres of corn and 20 of soybeans. Based on his calculations, he can earn $150 per acre of corn and $250 per acre of soybeans, how many acres of each should he plant to earn the greatest profit? What is the farmer's maximum profit?

This detailed solution demonstrates how to use linear programming to maximize the profit of a farmer growing corn and soybeans. Using constraints like total land availability and minimum crop acreage, learn how to calculate the maximum profit of $116,000 by optimizing the number of acres planted. Suitable for high school students studying optimization and linear programming.

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