13. The Point At Which The Maximum Value Of Z=3x+ 2y Subject To The Constraints X+2y≤2, X ≥0, Y ≥0 Is (a) (0, 0) (b) (1.5, 1.5) (c) (2, 0) (d) (0, 2)​

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Introduction

Linear programming is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships. In this article, we will explore how to find the maximum value of a linear function subject to certain constraints. We will use the given problem, Z = 3x + 2y, subject to the constraints x + 2y ≤ 2, x ≥ 0, and y ≥ 0, to demonstrate the process.

Understanding the Problem

The problem requires us to find the maximum value of the linear function Z = 3x + 2y, subject to the constraints x + 2y ≤ 2, x ≥ 0, and y ≥ 0. This is a classic example of a linear programming problem, where we need to optimize a linear objective function subject to linear constraints.

Graphical Method

To solve this problem, we can use the graphical method. We will first plot the constraint lines on a coordinate plane. The constraint x + 2y ≤ 2 can be written as y ≤ -1/2x + 1. We will plot this line on the coordinate plane.

# Graphical Method

## Plotting the Constraint Lines

The constraint x + 2y ≤ 2 can be written as y ≤ -1/2x + 1. We will plot this line on the coordinate plane.

### Plotting the Line y = -1/2x + 1

The line y = -1/2x + 1 is a downward-sloping line that intersects the y-axis at (0, 1) and the x-axis at (2, 0).

Finding the Feasible Region

The feasible region is the area on the coordinate plane that satisfies all the constraints. In this case, the feasible region is the area below the line y = -1/2x + 1 and above the x-axis.

# Finding the Feasible Region

## Plotting the Feasible Region

The feasible region is the area below the line y = -1/2x + 1 and above the x-axis.

### Plotting the Feasible Region

The feasible region is a triangular area with vertices at (0, 0), (2, 0), and (0, 1).

Finding the Optimal Solution

The optimal solution is the point in the feasible region that maximizes the objective function Z = 3x + 2y. To find the optimal solution, we can use the corner point method. We will evaluate the objective function at each corner point of the feasible region and choose the point that maximizes the objective function.

# Finding the Optimal Solution

## Evaluating the Objective Function at Each Corner Point

We will evaluate the objective function Z = 3x + 2y at each corner point of the feasible region.

### Evaluating the Objective Function at (0, 0)

Z = 3(0) + 2(0) = 0

### Evaluating the Objective Function at (2, 0)

Z = 3(2) + 2(0) = 6

### Evaluating the Objective Function at (0, 1)

Z = 3(0) + 2(1) = 2

### Choosing the Optimal Solution

The optimal solution is the point that maximizes the objective function. In this case, the optimal solution is (2, 0), which maximizes the objective function Z = 3x + 2y.

Conclusion

In this article, we used the graphical method to find the maximum value of the linear function Z = 3x + 2y, subject to the constraints x + 2y ≤ 2, x ≥ 0, and y ≥ 0. We plotted the constraint lines on a coordinate plane, found the feasible region, and evaluated the objective function at each corner point of the feasible region. We chose the point that maximizes the objective function as the optimal solution. The optimal solution is (2, 0), which maximizes the objective function Z = 3x + 2y.

Answer

Introduction

Linear programming is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships. In this article, we will explore some frequently asked questions about linear programming and provide answers to help you better understand the concept.

Q1: What is Linear Programming?

A1: Linear programming is a method to achieve the best outcome in a given mathematical model for some list of requirements represented as linear relationships. It is a technique used to optimize a linear objective function subject to linear constraints.

Q2: What are the Key Components of Linear Programming?

A2: The key components of linear programming are:

  • Objective Function: A linear function that needs to be optimized.
  • Constraints: Linear inequalities that restrict the feasible region.
  • Feasible Region: The area on the coordinate plane that satisfies all the constraints.
  • Optimal Solution: The point in the feasible region that maximizes or minimizes the objective function.

Q3: What are the Types of Linear Programming Problems?

A3: There are two types of linear programming problems:

  • Maximization Problem: The objective function needs to be maximized.
  • Minimization Problem: The objective function needs to be minimized.

Q4: How do I Solve a Linear Programming Problem?

A4: To solve a linear programming problem, you can use the following steps:

  1. Plot the Constraint Lines: Plot the constraint lines on a coordinate plane.
  2. Find the Feasible Region: Find the area on the coordinate plane that satisfies all the constraints.
  3. Evaluate the Objective Function: Evaluate the objective function at each corner point of the feasible region.
  4. Choose the Optimal Solution: Choose the point that maximizes or minimizes the objective function.

Q5: What are the Applications of Linear Programming?

A5: Linear programming has numerous applications in various fields, including:

  • Business: Linear programming is used to optimize production levels, inventory levels, and resource allocation.
  • Economics: Linear programming is used to optimize economic models, such as supply and demand models.
  • Engineering: Linear programming is used to optimize design and resource allocation in engineering projects.
  • Finance: Linear programming is used to optimize investment portfolios and risk management.

Q6: What are the Advantages of Linear Programming?

A6: The advantages of linear programming include:

  • Optimization: Linear programming can optimize a linear objective function subject to linear constraints.
  • Flexibility: Linear programming can handle a wide range of problems, from simple to complex.
  • Accuracy: Linear programming can provide accurate solutions to problems.

Q7: What are the Limitations of Linear Programming?

A7: The limitations of linear programming include:

  • Linearity: Linear programming assumes that the objective function and constraints are linear.
  • Scalability: Linear programming can become computationally intensive for large-scale problems.
  • Non-Convexity: Linear programming assumes that the feasible region is convex.

Conclusion

In this article, we answered some frequently asked questions about linear programming and provided answers to help you better understand the concept. Linear programming is a powerful tool for optimizing linear objective functions subject to linear constraints. It has numerous applications in various fields and can provide accurate solutions to problems. However, it also has some limitations, such as linearity and scalability.