Questions: Find the minimum and maximum P value for the following system. Minimize and Maximize P=30x+10y Subject to: 2x+2y ≥ 4 6x+4y ≤ 36 2x+y ≤ 10 x, y ≥ 0 The minimum value of P for this system is and occurs at The maximum value of P for this system is and occurs at

Find the minimum and maximum P value for the following system. Minimize and Maximize P=30x+10y Subject to: 2x+2y ≥ 4 6x+4y ≤ 36 2x+y ≤ 10 x, y ≥ 0 The minimum value of P for this system is and occurs at The maximum value of P for this system is and occurs at
Transcript text: Find the minimum and maximum P value for the following system. \[ \begin{array}{l} \text { Minimize and Maximize } P=30 x+10 y \\ \text { Subject to: } \quad \begin{array}{ll} 2 x+2 y & \geq 4 \\ 6 x+4 y \leq 36 \\ 2 x+y & \leq 10 \\ & x, y \geq 0 \end{array} \end{array} \] The minimum value of $P$ for this system is $\square$ and occurs at $\square$ The maximum value of $P$ for this system is $\square$ and occurs at $\square$
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Solution

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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 \( P = 30x + 10y \) at each vertex of this region. The minimum and maximum values of \( P \) will occur at one of these vertices. We will use Python's scipy.optimize.linprog to find these values.

Step 1: Define the Objective Function and Constraints

We are given the objective function to optimize: \[ P = 30x + 10y \]

The constraints for the system are: \[ \begin{align_} 2x + 2y & \geq 4 \\ 6x + 4y & \leq 36 \\ 2x + y & \leq 10 \\ x, y & \geq 0 \end{align_} \]

Step 2: Convert Inequalities for Linear Programming

To use linear programming, we convert the inequalities: \[ \begin{align_} 2x + 2y & \geq 4 \quad \Rightarrow \quad -2x - 2y \leq -4 \\ 6x + 4y & \leq 36 \\ 2x + y & \leq 10 \end{align_} \]

Step 3: Solve for Maximum and Minimum Values

Using the constraints, we find the feasible region and evaluate the objective function at the vertices of this region. The results are:

  • The minimum value of \( P \) is \(-20\) and occurs at the point \((0, 2)\).
  • The maximum value of \( P \) is \(150\) and occurs at the point \((5, 0)\).

Final Answer

The minimum value of \( P \) is \(\boxed{-20}\) and occurs at \(\boxed{(0, 2)}\).

The maximum value of \( P \) is \(\boxed{150}\) and occurs at \(\boxed{(5, 0)}\).

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