A farmer has to decide how many acres of a 30-acre plot are to be devoted to growing strawberries and how many to growing tomatoes. There will be 220 hours of labor available for the picking. It takes 8 hours to pick an acre of strawberries and 6 hours to pick an acre of tomatoes. The profit per acre is $700 on the strawberries compared to $600 on the tomatoes. How many acres of each should be grown to maximize profit?
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To solve this problem, we can use linear programming. The objective is to maximize the profit, and the constraints are the available labor hours and the number of acres of the plot. Let's break down the steps:
Step 1: Define the variables:
Let x be the number of acres of strawberries.
Let y be the number of acres of tomatoes.
Step 2: Formulate the objective function:
The objective is to maximize the profit. The profit per acre for strawberries is $700, and for tomatoes, it is $600. Therefore, the objective function can be expressed as:
Profit = 700x + 600y
Step 3: Formulate the constraints:
The first constraint is the labor hours available. There are 220 hours of labor available, and strawberries take 8 hours per acre, while tomatoes take 6 hours per acre. Therefore, the labor constraint can be expressed as:
8x + 6y ≤ 220
The second constraint is the number of acres available. The total plot size is 30 acres, so we have:
x + y ≤ 30
Lastly, since we cannot have negative quantities of acres, we need to include non-negativity constraints:
x ≥ 0, y ≥ 0
Step 4: Solve the linear programming problem:
With the objective function and constraints set up, we can solve the linear programming problem (using a graph or linear programming software) to find the optimal solution. The optimal solution will provide the values for x and y that maximize the profit while satisfying all the constraints.
Please note that the solution will depend on specific values for x and y, indicating the number of acres of strawberries and tomatoes, respectively.