Question

A farmer has 360 acres of land to plant corn and wheat and a $24,000 budget. To make 100 bushels of corn which is an acre is $120 and an acre of wheat which is 40 bushels is $60. The farmer can only have 18,000 bushels and wants to maximize profits. You can sell the 100bushells/acre of corn for $225 and an acre of wheat/40bushells for $100.

How many acres of corn and wheat should the farmer produce to maximize profits but not go over the 18,000 bushel limit?

Answer 1

To maximize the farmer's profits, we define the number of acres for corn and wheat as variables and apply linear programming considering constraints on land, budget, and bushel limit. We solve for the combination of corn and wheat that will result in the highest profit without exceeding any constraints.

Step-by-step explanation:

To solve this problem, we need to use linear programming to maximize the farmer's profit given the constraints. We define the variables as the number of acres of corn (x) and wheat (y). So, we have the following constraints based on land, budget, and bushel limits:

• x + y ≤ 360 (land constraint)
• 120x + 60y ≤ 24000 (budget constraint)
• 100x + 40y ≤ 18000 (bushel constraint)

The profit function to maximize is: Profit = 225x + 100y.

When we solve these linear inequalities, we aim to find the optimal combination of x and y. Graphically, we look for the point within the feasible region that will give us the highest value for the Profit function. We check the vertices of the feasible region which are formed at the intersection of constraints. The final solution will be where the maximum profit can be obtained without exceeding constraints. Using linear programming methods or graphical analysis, we would calculate the exact number of acres to plant for each crop.