### Solving a Linear Programming ProblemLinear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Here, we will solve a linear programming problem to maximize an objective function.#### Objective:Maximize $ P = 20x + 25y $#### Subject to the Constraints:\[ \begin{align*}0.6x + 1.2y & \leq 960 \\0.03x + 0.04y & \leq 36 \\0.3x + 0.2y & \leq 285 \\x, y & \geq 0 \end{align*}\]#### Example Problem:Let's consider the following problem for maximizing the value of $ P $:**Objective Function:**\[ P = 20x + 25y \]**Constraints:**\[ \begin{align*}0.6x + 1.2y & \leq 960 \\0.03x + 0.04y & \leq 36 \\0.3x + 0.2y & \leq 285 \\x, y & \geq 0 \end{align*}\]**Question:**What is the maximum value of $ P $?When solving such problems, one can use graphical methods for two-variable problems or simplex methods for more complex cases. The feasible region is determined by the intersection of the half-planes defined by the inequalities. The optimal solution will be located at one of the vertices (corner points) of the feasible region.

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Solve the linear programming problem. Maximize P=20x+25y Subject to 0.6x+1.2y s 960 0.03x+0.04y s 36 0.3x+0.2y s 285 X, y 2 0 What is the maximum value of P? Select th

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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### Solving a Linear Programming Problem

Linear programming is a method to achieve the best outcome in a mathematical model whose requirements are represented by linear relationships. Here, we will solve a linear programming problem to maximize an objective function.

#### Objective:
Maximize $ P = 20x + 25y $

#### Subject to the Constraints:
\[
\begin{align*}
0.6x + 1.2y & \leq 960 \\
0.03x + 0.04y & \leq 36 \\
0.3x + 0.2y & \leq 285 \\
x, y & \geq 0
\end{align*}
\]

#### Example Problem:
Let's consider the following problem for maximizing the value of $ P $:

**Objective Function:**
\[ P = 20x + 25y \]

**Constraints:**
\[
\begin{align*}
0.6x + 1.2y & \leq 960 \\
0.03x + 0.04y & \leq 36 \\
0.3x + 0.2y & \leq 285 \\
x, y & \geq 0
\end{align*}
\]

**Question:**
What is the maximum value of $ P $?

When solving such problems, one can use graphical methods for two-variable problems or simplex methods for more complex cases. The feasible region is determined by the intersection of the half-planes defined by the inequalities. The optimal solution will be located at one of the vertices (corner points) of the feasible region.

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