Maximize the object function $z = 4 x + 3 y$ subject to the constraints $3 x + y \leq 15,4 x + 3 y \leq 30 , x \geq 0 , \text { and } y \geq 0$ .
A)The maximum value is 24 along the line segment connection $( 0,10 ) \text { and } ( 3,4 )$ .
B)The maximum value is 30 along the line segment connection $( 0,15 ) \text { and } ( 3,6 )$ .
C)The maximum value is 30 along the line segment connection $( 0,10 ) \text { and } ( 3,6 )$ .
D)The maximum value is 24 along the line segment connection $( 0,15 ) \text { and } ( 3,4 )$ .
E)The maximum value is 30 along the line segment connection $( 0,30 ) \text { and } ( 3,6 )$ .

Question

Quizplus · Accepted Answer

To find the maximum value of the objective function $z = 4x + 3y$ subject to the given constraints, we can graph the constraints and find the feasible region. The vertices of this region are where the maximum or minimum values occur for linear programming problems. Evaluating $z$ at these vertices:1. At $x = 0, 3x + y \leq 15$ and $4x + 3y \leq 30$ become $y \leq 15$ and $3y \leq 30$, respectively. The intersection point here doesn't directly give a vertex, but evaluating $z$ at $y = 10$ (a point on the feasible boundary) gives $z = 4(0) + 3(10) = 30$.2. Solving the system of equations given by the constraints $3x + y = 15$ and $4x + 3y = 30$ can give another vertex. For simplicity, let's find an intersection point that could serve as a vertex. One such point is $(3, 6)$, which can be found by solving the equations simultaneously or by graphical analysis. At this point, $z = 4(3) + 3(6) = 12 + 18 = 30$.3. The points $x = 0, y = 10$ and $x = 3, y = 6$ are on the boundary of the feasible region and contribute to the maximum value of $z$, which is 30.Thus, the maximum value of the objective function is 30, and it occurs along the line segment connecting the points $(0, 10)$ and $(3, 6)$, making option C correct.

Maximize the Object Function z=4x+3yz = 4 x + 3 yz=4x+3y

Maximize the Object Function $z = 4 x + 3 y$