QQuestionMathematics
QuestionMathematics
# Exercise 7.4
Solve the following problems:
## Objective <br> Constrain
1. Maximize $z= 0.8 x- 0.5 y$
$x-y \geq- 2$
$2 x-y \leq 4$
$2 x+y \leq 8$
$x, y \geq 0$
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Answer
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Step 1:: Graph the constraints and identify the feasible region.
First, let's rewrite the constraints in slope-intercept form to graph them:
Step 2:$y \geq x + 2$
2. $y \leq 2x - 4$
Step 3:$y \leq - 2x + 8$
4. $y \geq 0$ and $x \geq 0$
Graph these lines and shade the corresponding regions. The feasible region is the area that satisfies all the constraints.
Step 4:: Find the corner points of the feasible region.
D: $(1, 3)$
The corner points are the points where the lines intersect. By solving the systems of equations, we find the following corner points:
Step 5:: Evaluate the objective function at each corner point.
D: $z(1, 3) = 1.4$
Step 6:: Identify the maximum value of the objective function.
The maximum value of the objective function is 1.6, which occurs at the corner point B: $(2, 0)$.
Final Answer
The maximum value of the objective function is 1.6, which occurs at the point $(2, 0)$.
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