Question

Maximize p = 6x 4y subject to x 3y ≥ 6 −x y ≤ 4 2x y ≤ 8 x ≥ 0, y ≥ 0.

Answer 1

To solve the linear programming problem, graph the feasible region and evaluate the objective function at each corner point.

Step-by-step explanation:

To solve the given linear programming problem, we need to graph the feasible region determined by the system of inequalities. Then, we can evaluate the objective function at each corner point of the feasible region to find the maximum value.

First, graph the lines x + 3y = 6, -x + y ≤ 4, and 2xy ≤ 8. The feasible region is the intersection of the shaded areas. The corner points of the feasible region are (0, 2), (2, 2), and (4, 0).

Next, evaluate the objective function p = 6x + 4y at each corner point and determine the maximum value. p(0, 2) = 12, p(2, 2) = 20, and p(4, 0) = 24. Therefore, the maximum value of p is 24, which occurs at (4, 0).

Answer 2

The question involves solving a linear programming problem by graphing inequalities and finding the maximum of the objective function p = 6x + 4y, subject to given constraints and non-negativity conditions.

Step-by-step explanation:

The student is asking about a problem in linear programming, which is an optimization technique used in mathematics to maximize or minimize a linear objective function, subject to a set of linear inequalities or constraints. The objective function given is p = 6x + 4y, and the constraints are x + 3y ≥ 6, −x + y ≤ 4, 2x + y ≤ 8, with x ≥ 0, y ≥ 0. To solve this, one must graph the inequalities to form a feasible region and then evaluate the objective function at each vertex of this region to find the maximum value of p.

Identifying the constraints and objective functions is essential for setting up the linear programming model. The independent and dependent variables are determined by the context of the problem, similar to how the year could be independent while flu cases are dependent in health-related data analysis. The variables x and y are typically labeled on the axes of the graph, which should also be scaled appropriately to accommodate the feasible region.