Questions: Solve the linear programming problem. Maximize P=4x+4y Subject to 2x+y ≤ 20 x+2y ≤ 16 x, y ≥ 0 Select the correct choice below and fill in any answer boxes present in your choice. What is the maximum value of P ? A. P= (Type an integer or a fraction.) B. There is no maximum value of P.

$Solve the linear programming problem. Maximize P=4x+4y Subject to 2x+y ≤ 20 x+2y ≤ 16 x, y ≥ 0 Select the correct choice below and fill in any answer boxes present in your choice. What is the maximum value of P ? A. P= (Type an integer or a fraction.) B. There is no maximum value of P.$

Transcript text: Solve the linear programming problem. Maximize \[ P=4 x+4 y \] Subject to \[ \begin{aligned} 2 x+y & \leq 20 \\ x+2 y & \leq 16 \\ x, y & \geq 0 \end{aligned} \] Select the correct choice below and fill in any answer boxes present in your choice. What is the maximum value of $P$ ? A. $\mathrm{P}=$ $\square$ (Type an integer or a fraction.) B. There is no maximum value of $P$.

Solution

Solution Steps

Step 1: Define the Objective Function

We want to maximize the objective function given by

\[ P = 4x + 4y. \]

Step 2: Identify the Constraints

The constraints for the problem are:

\[ \begin{aligned} 2x + y & \leq 20, \\ x + 2y & \leq 16, \\ x, y & \geq 0. \end{aligned} \]

Step 3: Determine the Feasible Region

The feasible region is defined by the intersection of the constraints. We can find the vertices of this region by solving the equations formed by the constraints.

Step 4: Evaluate the Objective Function at the Vertices

After determining the vertices of the feasible region, we evaluate the objective function $P$ at each vertex. The maximum value of $P$ occurs at one of these vertices.

Step 5: Calculate the Maximum Value

The maximum value of $P$ found at the vertices is

\[ \max P = 48.0. \]