A farmer has 110 acres available for planting cucumbers and...

A farmer has 110 acres available for planting cucumbers and peanuts. The cost of seed per acre is $\$ 5$ for cucumbers and $\$ 6$ for peanuts. To harvest the crops, the farmer will need to hire some temporary help. It will cost the farmer $\$ 30$ per acre to harvest the cucumbers and $\$ 20$ per acre to harvest the peanuts. The farmer has $\$ 300$ available for seed and $\$ 1200$ available for labor. His profit is $\$ 100$ per acre of cucumbers and $\$ 125$ per acre of peanuts. How many acres of each crop should the farmer plant to maximize the profit?

Key Concepts

Feasible Region

The feasible region in a linear programming problem is the set of all points that satisfy the given constraints. It is the space within which the decision variables can vary, and the optimal solution lies at one of the vertices or along the boundary of this region.

Constraints

Constraints are the conditions or restrictions that the decision variables must satisfy in an optimization problem. They model the resource limitations and other requirements, such as available acreage, seed costs, and labor costs, ensuring that any solution is feasible within the provided limits.

Graphical Method

The graphical method is a technique used to solve linear programming problems with two decision variables by plotting the constraints on a graph. It helps to visually identify the feasible region and determine the optimal solution by evaluating the objective function at the vertices of this region.

Decision Variables

Decision variables represent the quantities that can be controlled or chosen in the problem. In optimization problems, these variables are used to formulate the quantities of interest, such as how many acres to allocate to different crops, allowing the problem to be expressed in mathematical terms.

Linear Programming

Linear programming is a mathematical optimization method used for decision-making in which a linear objective function is maximized or minimized subject to a set of linear equality or inequality constraints. It is widely used in various fields, including agriculture, to allocate limited resources efficiently.

Objective Function

The objective function is a mathematical expression representing the goal of the optimization problem, such as maximizing profit or minimizing cost. It is formulated as a linear combination of decision variables and serves as the criterion to evaluate potential solutions.

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