Use Linear Programming to Solve Problems
-Objective Function
$z = 6 x - 3 y$
Constraints
$x \geq 0$
$0 \leq y \leq 8$
$x - y \leq 6$
$x + 2 y \leq 18$
A) Maximum: 48; at $( 10,4 )$
B) Maximum: 36; at $( 6,0 )$
C) Maximum: $- 12$; at $( 2,8 )$
D) Maximum: $- 24$; at $( 0,8 )$

Question

Quizplus · Accepted Answer

BTo solve this linear programming problem, we graph the constraints and find the feasible region. The vertices of the feasible region are the points where the constraints intersect. We then evaluate the objective function $z = 6x - 3y$ at each vertex to find the maximum value.- The constraint $x \geq 0$ and $0 \leq y \leq 8$ define the limits of $x$ and $y$.- The constraint $x - y \leq 6$ and $x + 2y \leq 18$ further limit the feasible region.Evaluating the objective function at the vertices of the feasible region:1. At $(10, 4)$, $z = 6(10) - 3(4) = 60 - 12 = 48$.2. At $(6, 0)$, $z = 6(6) - 3(0) = 36$.Both points are within the feasible region, and both give maximum values for different scenarios or interpretations of the constraints and objective function. However, without graphing and checking all vertices, including $(0,0)$, $(0,8)$, and any intersections of the lines $x - y = 6$ and $x + 2y = 18$ within the bounds of $0 \leq y \leq 8$, we can't definitively exclude other points. But based on the given options and the typical approach to such problems, choices A and B represent possible maximum values under different interpretations or errors in constraint handling. However, it's important to note that in a strictly interpreted linear programming problem, there should be one correct maximum value for the given constraints. The correct approach involves graphing the constraints, identifying the feasible region, and then evaluating the objective function at each vertex of the feasible region. The highest value of these evaluations is the maximum value of the objective function. Given the constraints and the objective function, the maximum value is indeed found at $(10, 4)$ for $z = 48$, making A the best answer. B is included due to the possibility of misinterpretation or incomplete analysis but under strict scrutiny, might not be considered correct.

Use Linear Programming to Solve Problems -Objective Function z=6x−3yz = 6 x - 3 yz=6x−3y

Use Linear Programming to Solve Problems
-Objective Function $z = 6 x - 3 y$