Question

An objective function and a system of linear inequalities representing constraints are given. a. Graph the system of inequalities representing the constraints. b. Find the value of the objective function at each corner of the graphed region. c. Use the values in part (b) to determine the maximum value of the objective function and the values of x and y for which the maximum occurs.

Accepted Answer

Step 1: Identify the objective function and the system of inequalities. The objective function is given by $$z = x + 8y. $$The constraints are: $$x \geq 0$$, $$y \geq 0$$, $$6x + 5y \geq 30$$, and $$4x + 5y \leq 40. $$Step 2: Graph the inequalities on the coordinate plane. Start by graphing the boundary lines for each inequality: $$x = 0$$, $$y = 0$$, $$6x + 5y = 30$$, and $$4x + 5y = 40. $$Use these lines to determine the feasible region that satisfies all inequalities simultaneously. Step 3: Find the corner points (vertices) of the feasible region by solving the systems of equations formed by the intersection of the boundary lines. For example, find the intersection of $$6x + 5y = 30$$ and $$4x + 5y = 40$$, as well as intersections with the axes where $$x=0 or$$ $$y=0. $$Step 4: Evaluate the objective function $$z = x + 8y at $$each corner point found in Step 3. Substitute the $$x$$ and $$y$$ values of each vertex into the objective function to find the corresponding $$z$$ values. Step 5: Compare the values of $$z$$ obtained at each corner point to determine the maximum value. Identify the corner point $$(x, y)$$ where this maximum occurs, which will be the solution to the optimization problem.

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Key Concepts

Graphing Systems of Linear Inequalities

Feasible Region and Corner Points

Evaluating the Objective Function at Corner Points