Final answer:
To maximize profit, the farmer should plant 15 acres of crop A and 25 acres of crop B. The total profit from crop A will be $1950, and from crop B, it will be $7750.
Step-by-step explanation:
The student is asking how to maximize profit by dividing a 40-acre farm between two crops with different costs and profit margins. Let's denote the number of acres planted with crop A as x and with crop B as y. The constraints are given by the seed costs and total acreage available. The objective function to maximize is the total profit.
- Seed for crop A costs $10 per acre, and for crop B, it costs $20 per acre. The total cost should not exceed $500: 10x + 20y ≤ 500.
- The total acreage for crops A and B can't exceed 40 acres: x + y ≤ 40.
- Crop A brings in a profit of $130 per acre, and crop B brings in a profit of $310 per acre. The profit from each crop is given by: 130x + 310y.
To maximize profit, let's find the combination of x and y that satisfies both constraints and maximizes the profit function. Since the farmer can spend up to $500 on seeds, the maximum number of acres that can be planted with crop B (the more expensive seed) is 25 acres (y = $500/$20). Planting more than 25 acres of crop B would exceed the budget. Therefore, the remaining 15 acres would be planted with crop A. This combination uses the entire budget with the following results:
- Total acres of crop A: 15 acres
- Total acres of crop B: 25 acres
- Total profit from crop A: 15 acres * $130/acre = $1950
- Total profit from crop B: 25 acres * $310/acre = $7750