A farmer has 90 acres available for planting corn and...

A farmer has 90 acres available for planting corn and soybeans. The cost of seed per acre is $\$ 4$ for corn and $\$ 6$ for soybeans. To harvest the crops, the farmer will need to hire some temporary help. It will cost the farmer $\$ 20$ per acre to harvest the corn and $\$ 10$ per acre to harvest the soybeans. The farmer has $\$ 480$ available for seed and $\$ 1400$ available for labor. His profit is $\$ 120$ per acre of corn and $\$ 150$ per acre of soybeans. How many acres of each crop should the farmer plant to maximize the profit?

A farmer has 90 acres available for planting corn and soybeans. The cost of seed per acre is
$\$ 4$ for corn and $\$ 6$ for soybeans. To harvest the crops, the farmer will need to hire some temporary help. It will cost the farmer $\$ 20$ per acre to harvest the corn and $\$ 10$ per acre to harvest the soybeans. The farmer has $\$ 480$ available for seed and $\$ 1400$ available for labor. His profit is $\$ 120$ per acre of corn and $\$ 150$ per acre of soybeans. How many acres of each crop should the farmer plant to maximize the profit?

Key Concepts

Linear Programming

Linear programming is a mathematical method used to determine the best possible outcome in a given mathematical model whose requirements are represented by linear relationships. It involves optimizing a linear objective function, subject to a set of linear equality and inequality constraints, and is widely used in various fields such as economics, business, engineering, and military planning.

Decision Variables

Decision variables are the unknowns in a linear programming problem that represent the quantities to be determined. These variables are used to formulate the objective function and constraints, and their optimal values yield the best solution with respect to the problem's objective.

Constraints

Constraints in linear programming are linear equations or inequalities that represent the limitations or requirements of the problem, such as resource limitations or minimum/maximum conditions. They define the boundaries of the solution space within which the optimal solution must be found.

Objective Function

The objective function is a linear function that represents the goal of the optimization problem, which is typically to maximize profit or minimize cost. It is constructed using decision variables and reflects the performance measure that needs to be optimized under the given constraints.

Feasible Region

The feasible region is the set of all possible values of the decision variables that satisfy all the constraints of a linear programming problem. It is typically represented graphically in two dimensions as a polygonal area, where every point within the region is a potential solution that meets all restrictions.

Corner Point Method

The corner point method is an approach used in linear programming, particularly for problems involving two variables, where the optimal solution is found at one of the vertices (corner points) of the feasible region. By evaluating the objective function at each corner point, one can identify which vertex gives the optimal result.

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