What Is The Maximum Value Of The Objective Function, $P$, With The Given Constraints?$P = 20x + 35y$Subject To:$\[ \begin{cases} x + Y \leq 12 \\ 5x + Y \leq 20 \\ x \geq 0 \\ y \geq 0 \end{cases} \\]A. 80 B. 390 C. 420

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Introduction

In linear programming, the objective function is a mathematical expression that represents the goal of the problem. The constraints are the limitations or restrictions that the solution must satisfy. In this article, we will explore the maximum value of the objective function, P, with the given constraints. The objective function is P = 20x + 35y, and the constraints are x + y ≤ 12, 5x + y ≤ 20, x ≥ 0, and y ≥ 0.

Understanding the Constraints

To find the maximum value of the objective function, we need to understand the constraints. The constraints are:

  • x + y ≤ 12
  • 5x + y ≤ 20
  • x ≥ 0
  • y ≥ 0

These constraints represent the limitations on the values of x and y. The first constraint, x + y ≤ 12, means that the sum of x and y must be less than or equal to 12. The second constraint, 5x + y ≤ 20, means that the sum of 5x and y must be less than or equal to 20. The third and fourth constraints, x ≥ 0 and y ≥ 0, mean that x and y must be non-negative.

Graphing the Constraints

To visualize the constraints, we can graph them on a coordinate plane. The first constraint, x + y ≤ 12, can be graphed as a line with a slope of -1 and a y-intercept of 12. The second constraint, 5x + y ≤ 20, can be graphed as a line with a slope of -5 and a y-intercept of 20. The third and fourth constraints, x ≥ 0 and y ≥ 0, can be graphed as the x-axis and y-axis, respectively.

Finding the Feasible Region

The feasible region is the area where all the constraints are satisfied. To find the feasible region, we need to graph the constraints and identify the area where all the constraints are satisfied. The feasible region is a polygon with vertices at (0, 0), (0, 12), (4, 0), and (0, 8).

Finding the Maximum Value of the Objective Function

To find the maximum value of the objective function, we need to find the point in the feasible region that maximizes the objective function. We can use the corner point method to find the maximum value of the objective function. The corner points of the feasible region are (0, 0), (0, 12), (4, 0), and (0, 8). We can evaluate the objective function at each of these points to find the maximum value.

Evaluating the Objective Function at Each Corner Point

We can evaluate the objective function at each of the corner points to find the maximum value.

  • At (0, 0), P = 20(0) + 35(0) = 0
  • At (0, 12), P = 20(0) + 35(12) = 420
  • At (4, 0), P = 20(4) + 35(0) = 80
  • At (0, 8), P = 20(0) + 35(8) = 280

Conclusion

The maximum value of the objective function, P, with the given constraints is 420. This occurs at the point (0, 12) in the feasible region.

Final Answer

The final answer is 420.

Introduction

In our previous article, we explored the maximum value of the objective function, P, with the given constraints. The objective function is P = 20x + 35y, and the constraints are x + y ≤ 12, 5x + y ≤ 20, x ≥ 0, and y ≥ 0. In this article, we will answer some frequently asked questions related to the maximum value of the objective function, P.

Q1: What is the objective function, P?

A1: The objective function, P, is a mathematical expression that represents the goal of the problem. In this case, the objective function is P = 20x + 35y.

Q2: What are the constraints?

A2: The constraints are the limitations or restrictions that the solution must satisfy. In this case, the constraints are x + y ≤ 12, 5x + y ≤ 20, x ≥ 0, and y ≥ 0.

Q3: How do we find the maximum value of the objective function, P?

A3: To find the maximum value of the objective function, P, we need to find the point in the feasible region that maximizes the objective function. We can use the corner point method to find the maximum value of the objective function.

Q4: What is the feasible region?

A4: The feasible region is the area where all the constraints are satisfied. In this case, the feasible region is a polygon with vertices at (0, 0), (0, 12), (4, 0), and (0, 8).

Q5: How do we evaluate the objective function at each corner point?

A5: We can evaluate the objective function at each of the corner points to find the maximum value. We can plug in the values of x and y at each corner point into the objective function and calculate the value of P.

Q6: What is the maximum value of the objective function, P?

A6: The maximum value of the objective function, P, is 420. This occurs at the point (0, 12) in the feasible region.

Q7: Why is the maximum value of the objective function, P, important?

A7: The maximum value of the objective function, P, is important because it represents the optimal solution to the problem. In this case, the maximum value of the objective function, P, is 420, which means that the optimal solution is to have x = 0 and y = 12.

Q8: Can we use other methods to find the maximum value of the objective function, P?

A8: Yes, we can use other methods to find the maximum value of the objective function, P. Some other methods include the graphical method, the simplex method, and the dual simplex method.

Q9: What are some common applications of linear programming?

A9: Some common applications of linear programming include resource allocation, production planning, and scheduling. Linear programming is also used in finance, economics, and engineering.

Q10: Can you provide some examples of real-world problems that can be solved using linear programming?

A10: Yes, some examples of real-world problems that can be solved using linear programming include:

  • A company wants to produce two products, A and B, using two resources, labor and capital. The company wants to maximize its profit, which is given by the objective function P = 20x + 35y, where x is the number of units of product A produced and y is the number of units of product B produced. The constraints are x + y ≤ 12, 5x + y ≤ 20, x ≥ 0, and y ≥ 0.
  • A farmer wants to plant two crops, wheat and corn, on a 100-acre farm. The farmer wants to maximize the total revenue, which is given by the objective function P = 20x + 35y, where x is the number of acres of wheat planted and y is the number of acres of corn planted. The constraints are x + y ≤ 100, 2x + y ≤ 80, x ≥ 0, and y ≥ 0.

Conclusion

In this article, we answered some frequently asked questions related to the maximum value of the objective function, P. We also provided some examples of real-world problems that can be solved using linear programming.