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 $$z = 2x + 6y$$, and the constraints are:

$$
\begin{cases}
x \geq 0, \
y \geq 0, \
-3x + 4y \leq 20, \
4x + y \leq 24, \
x + 2y \geq 4
\end{cases}
$$ Step 2: Graph the inequalities one by one on the coordinate plane. Start by graphing the boundary lines (replace inequalities with equalities):

- $$x = 0 ($$the y-axis),
- $$y = 0 ($$the x-axis),
- $$-3x + 4y = 20$$,
- $$4x + y = 24$$,
- $$x + 2y = 4.

$$For each line, determine which side satisfies the inequality by testing a point not on the line (usually the origin if it is not on the line). Step 3: Determine the feasible region by finding the intersection of all the half-planes defined by the inequalities. This region will be bounded by the lines and will satisfy all constraints simultaneously. Step 4: 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, solve pairs like:

- $$x=0$$ and $$-3x + 4y = 20$$,
- $$-3x + 4y = 20$$ and $$4x + y = 24$$,
- $$4x + y = 24$$ and $$x + 2y = 4$$,
- $$x + 2y = 4$$ and $$x=0$$,

and any other relevant intersections that form the polygon of the feasible region. Step 5: Evaluate the objective function $$z = 2x + 6y at $$each corner point found in Step 4. Compare these values to determine which corner point gives the maximum value of $$z. $$The maximum value and the corresponding $$(x, y)$$ coordinates are the solution to the problem.

Risposta video verificata per un problema simile:

Concetti chiave

Graphing Systems of Linear Inequalities

Corner Points (Vertices) of the Feasible Region

Evaluating the Objective Function at Vertices