A telephone company manufactures two different models of...

A telephone company manufactures two different models of phones: Model 120 is cordless and Model 140 is not cordless. It takes 1 hour to manufacture the cordless model and 1 hour and 30 minutes to manufacture the traditional phone. At least 300 of the cordless models are to be produced. The manufacturer realizes a profit per phone of $\$ 12$ for Model 120 and $\$ 10$ for Model $140 .$ If at most 1000 hours are to be allocated to the manufacture of the two models combined, how many of each model should be produced to maximize the total profit?

Key Concepts

Linear Programming

Linear programming is a method used in optimization that involves maximizing or minimizing a linear objective function, subject to a set of linear equality and inequality constraints. It is widely applied in various fields, including production planning, to determine the best outcomes in resource allocation problems.

Decision Variables

Decision variables in linear programming represent the quantities to be determined, such as the number of units to produce. They are the foundation of the model, as they translate the problem's requirements into mathematical terms.

Objective Function

The objective function in a linear programming problem defines the goal of the optimization, such as maximizing profit or minimizing cost. It is a linear expression of the decision variables that needs to be optimized subject to the constraints.

Constraints

Constraints are the restrictions or limitations imposed on the decision variables, expressed as linear inequalities or equalities. These could include resource limitations, production requirements, or time restrictions, and they define the feasible limits within which the decision variables must operate.

Feasible Region

The feasible region is the set of all possible solutions that satisfy all the constraints of the linear programming problem. It is typically represented graphically as a convex polygon, where the optimal solution, according to the Corner Point Theorem, lies at one of the vertices (corner points).

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