Questions: Use graphical methods to solve this linear programming problem. Maximize z=6x+3y subject to: 5x-y ≤ 16 3x+y ≥ 14 x ≥ 3 y ≤ 8 What is the maximum value of z? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The maximum value is . (Simplify your answer.) B. There is no maximum.

Use graphical methods to solve this linear programming problem. Maximize z=6x+3y subject to: 5x-y ≤ 16 3x+y ≥ 14 x ≥ 3 y ≤ 8 What is the maximum value of z? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The maximum value is . (Simplify your answer.) B. There is no maximum.
Transcript text: Use graphical methods to solve this linear programming problem. \[ \begin{array}{ll} \text { Maximize } & z=6 x+3 y \\ \text { subject to: } & 5 x-y \leq 16 \\ & 3 x+y \geq 14 \\ & x \geq 3 \\ & y \leq 8 \end{array} \] What is the maximum value of $z$ ? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The maximum value is $\square$ . (Simplify your answer.) B. There is no maximum.
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Solution

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Solution Steps

Step 1: Identify the Constraints and Objective Function

We are given the following linear programming problem:

Maximize \( z = 6x + 3y \)

Subject to: \[ \begin{cases} 5x - y \leq 16 \\ 3x + y \geq 14 \\ x \geq 3 \\ y \leq 8 \end{cases} \]

Step 2: Convert Inequalities to Equations

To graph the constraints, we first convert the inequalities to equations:

  1. \( 5x - y = 16 \)
  2. \( 3x + y = 14 \)
  3. \( x = 3 \)
  4. \( y = 8 \)
Step 3: Graph the Constraints

We will plot these lines on a graph to find the feasible region.

  1. For \( 5x - y = 16 \):

    • When \( x = 0 \), \( y = -16 \)
    • When \( y = 0 \), \( x = \frac{16}{5} = 3.2 \)

  2. For \( 3x + y = 14 \):

    • When \( x = 0 \), \( y = 14 \)
    • When \( y = 0 \), \( x = \frac{14}{3} \approx 4.67 \)

  3. For \( x = 3 \):

    • This is a vertical line at \( x = 3 \).

  4. For \( y = 8 \):

    • This is a horizontal line at \( y = 8 \).
Step 4: Determine the Feasible Region

The feasible region is the area where all the constraints overlap. We need to find the intersection points of the lines to determine the vertices of the feasible region.

Step 5: Find Intersection Points

  1. Intersection of \( 5x - y = 16 \) and \( 3x + y = 14 \): \[ \begin{cases} 5x - y = 16 \\ 3x + y = 14 \end{cases} \] Adding the equations: \[ 8x = 30 \implies x = \frac{30}{8} = 3.75 \] Substituting \( x = 3.75 \) into \( 3x + y = 14 \): \[ 3(3.75) + y = 14 \implies 11.25 + y = 14 \implies y = 2.75 \] So, the intersection point is \( (3.75, 2.75) \).

  2. Intersection of \( 5x - y = 16 \) and \( x = 3 \): \[ 5(3) - y = 16 \implies 15 - y = 16 \implies y = -1 \] So, the intersection point is \( (3, -1) \).

  3. Intersection of \( 3x + y = 14 \) and \( x = 3 \): \[ 3(3) + y = 14 \implies 9 + y = 14 \implies y = 5 \] So, the intersection point is \( (3, 5) \).

  4. Intersection of \( y = 8 \) and \( x = 3 \): \[ (3, 8) \]

Step 6: Evaluate the Objective Function at Each Vertex

We evaluate \( z = 6x + 3y \) at each vertex of the feasible region:

  1. At \( (3.75, 2.75) \): \[ z = 6(3.75) + 3(2.75) = 22.5 + 8.25 = 30.75 \]

  2. At \( (3, -1) \): \[ z = 6(3) + 3(-1) = 18 - 3 = 15 \]

  3. At \( (3, 5) \): \[ z = 6(3) + 3(5) = 18 + 15 = 33 \]

  4. At \( (3, 8) \): \[ z = 6(3) + 3(8) = 18 + 24 = 42 \]

Final Answer

The maximum value of \( z \) is at the point \( (3, 8) \).

\[ \boxed{z = 42} \]

Thus, the answer is A. The maximum value is \( 42 \).

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