Question

Your farm encompasses 900 acres, and you are planning to grow soybeans, corn, and wheat in the coming planting season. Fertilizer costs per acre are: $5 for soybeans, $2 for corn, and $1 for wheat. You estimate that each acre of soybeans will require an average of 5 hours of labor per week, while tending to corn and wheat will each require an average of 2 hours per week. Based on past yields and current market prices, you estimate a profit of $9,000 for each acre of soybeans, $6,000 for each acre of corn, and $3,000 for each acre of wheat. You can afford to spend no more than $5,400 on fertilizer, and your farm laborers can supply 5,400 hours per week. How many acres of each crop should you plant to maximize total profits

Answer 1

The problem can be solved using a linear programming approach to maximize profit, considering constraints on acreage, fertilizer costs, and labor hours. The objective is to maximize the profit function, given the constraints on the number of acres, cost of fertilizer, and available labor.

Step-by-step explanation:

To determine how many acres of soybeans, corn, and wheat should be planted to maximize profits under the given constraints, we'll need to set up a linear programming problem. We have the following conditions:

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Let's designate the number of acres for soybeans, corn, and wheat as S, C, and W, respectively. The linear programming problem can be formulated as follows:

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Using these equations, we can solve this problem either graphically or by using linear programming software to find the values of S, C, and W that will yield the maximum profit while satisfying all constraints.