Questions: Solve the linear programming problem Minimize and maximize z=40 x+50 y Subject to 3 x+y >= 48 x+y >= 24 x+3 y >= 36 x, y >= 0 Select the correct choice below and fill in any answer boxes present in your choice. What is the minimum value of z ? A. z=1020 (Type an integer or a fraction.) B. There is no minimum value of z. Select the correct choice below and fill in any answer boxes present in your choice. What are the coordinates of the corner point where the minimum value of z occurs? A. The coordinates are (18,6) (Type an ordered pair.) B. There is no minimum value of z Select the correct choice below and fill in any answer boxes present in your choice. What is the maximum value of z ? A. z= (Type an integer or a fraction.) B. There is no maximum value of z.

Solution Steps

To solve this linear programming problem, we need to find the feasible region defined by the constraints and then evaluate the objective function \( z = 40x + 50y \) at each corner point of this region. The minimum and maximum values of \( z \) will occur at one of these corner points. We will use Python's scipy.optimize.linprog to find these values.

We are given a linear programming problem where we need to minimize and maximize the objective function: \[ z = 40x + 50y \]

Subject to the constraints: \[ \begin{aligned} 3x + y & \geq 48 \\ x + y & \geq 24 \\ x + 3y & \geq 36 \\ x, y & \geq 0 \end{aligned} \]

To find the feasible region, we need to graph the constraints. We will convert each inequality to an equation to find the boundary lines.

1. \(3x + y = 48\)
2. \(x + y = 24\)
3. \(x + 3y = 36\)

Solve the system of equations to find the intersection points of the boundary lines.

1. Intersection of \(3x + y = 48\) and \(x + y = 24\):

   Subtract the second equation from the first: \[ (3x + y) - (x + y) = 48 - 24 \implies 2x = 24 \implies x = 12 \] Substitute \(x = 12\) into \(x + y = 24\): \[ 12 + y = 24 \implies y = 12 \] Intersection point: \((12, 12)\)

2. Intersection of \(x + y = 24\) and \(x + 3y = 36\):

   Subtract the first equation from the second: \[ (x + 3y) - (x + y) = 36 - 24 \implies 2y = 12 \implies y = 6 \] Substitute \(y = 6\) into \(x + y = 24\): \[ x + 6 = 24 \implies x = 18 \] Intersection point: \((18, 6)\)

3. Intersection of \(3x + y = 48\) and \(x + 3y = 36\):

   Multiply the second equation by 3: \[ 3(x + 3y) = 3 \times 36 \implies 3x + 9y = 108 \] Subtract the first equation from this result: \[ (3x + 9y) - (3x + y) = 108 - 48 \implies 8y = 60 \implies y = 7.5 \] Substitute \(y = 7.5\) into \(3x + y = 48\): \[ 3x + 7.5 = 48 \implies 3x = 40.5 \implies x = 13.5 \] Intersection point: \((13.5, 7.5)\)

Calculate \(z = 40x + 50y\) at each intersection point:

1. At \((12, 12)\): \[ z = 40(12) + 50(12) = 480 + 600 = 1080 \]

2. At \((18, 6)\): \[ z = 40(18) + 50(6) = 720 + 300 = 1020 \]

3. At \((13.5, 7.5)\): \[ z = 40(13.5) + 50(7.5) = 540 + 375 = 915 \]

• The minimum value of \(z\) is 915 at the point \((13.5, 7.5)\).
• The maximum value of \(z\) is 1080 at the point \((12, 12)\).

Final Answer