The Vertices Of A Feasible Region Are \[$(14,2)\$\], \[$(0,9)\$\], \[$(6,8)\$\], And \[$(10,3)\$\].What Is The Maximum Value Of The Function \[$P\$\] If \[$P = 180x + 250y\$\]?A. 2,940 B. 3,020 C. 3,080

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Introduction

In linear programming, the feasible region is the set of all possible solutions to a system of linear inequalities. The vertices of the feasible region are the points where the boundary lines intersect, and these points are critical in determining the maximum or minimum value of a linear function. In this article, we will explore how to find the maximum value of a linear function in a feasible region using the given vertices.

Understanding the Problem

The problem provides the vertices of a feasible region as {(14,2)$}$, {(0,9)$}$, {(6,8)$}$, and {(10,3)$}$. We are also given a linear function {P = 180x + 250y$}$ and asked to find the maximum value of {P$}$ in the feasible region.

The Linear Function

A linear function is a function that can be written in the form {f(x) = ax + b$}$, where {a$}$ and {b$}$ are constants. In this case, the linear function is {P = 180x + 250y$}$, where {x$}$ and {y$}$ are the variables.

Evaluating the Function at the Vertices

To find the maximum value of the function {P$}$ in the feasible region, we need to evaluate the function at each of the vertices. This involves substituting the coordinates of each vertex into the function and calculating the resulting value.

Evaluating the Function at {(14,2)$}$

Substituting {x = 14$}$ and {y = 2$}$ into the function {P = 180x + 250y$}$, we get:

{P = 180(14) + 250(2)$]

[P = 2520 + 500\$}

{P = 3020$}$

Evaluating the Function at {(0,9)$}$

Substituting {x = 0$}$ and {y = 9$}$ into the function {P = 180x + 250y$}$, we get:

{P = 180(0) + 250(9)$]

[P = 0 + 2250\$}

{P = 2250$}$

Evaluating the Function at {(6,8)$}$

Substituting {x = 6$}$ and {y = 8$}$ into the function {P = 180x + 250y$}$, we get:

{P = 180(6) + 250(8)$]

[P = 1080 + 2000\$}

{P = 3080$}$

Evaluating the Function at {(10,3)$}$

Substituting {x = 10$}$ and {y = 3$}$ into the function {P = 180x + 250y$}$, we get:

{P = 180(10) + 250(3)$]

[P = 1800 + 750\$}

{P = 2550$}$

Conclusion

In conclusion, we have evaluated the linear function {P = 180x + 250y$}$ at each of the vertices of the feasible region and found the maximum value of {P$}$ to be ${3080\$}. This is the highest value of {P$}$ among the four vertices, and therefore, it is the maximum value of the function in the feasible region.

Answer

Introduction

In our previous article, we explored how to find the maximum value of a linear function in a feasible region using the given vertices. In this article, we will answer some frequently asked questions related to the topic.

Q&A

Q: What is a feasible region?

A: A feasible region is the set of all possible solutions to a system of linear inequalities. It is the area where the constraints of the problem are satisfied.

Q: What are the vertices of a feasible region?

A: The vertices of a feasible region are the points where the boundary lines intersect. These points are critical in determining the maximum or minimum value of a linear function.

Q: How do I find the maximum value of a linear function in a feasible region?

A: To find the maximum value of a linear function in a feasible region, you need to evaluate the function at each of the vertices of the feasible region. This involves substituting the coordinates of each vertex into the function and calculating the resulting value.

Q: What is the significance of the vertices in finding the maximum value of a linear function?

A: The vertices are significant because they represent the extreme points of the feasible region. The maximum value of a linear function will occur at one of these vertices.

Q: Can the maximum value of a linear function occur at a point inside the feasible region?

A: No, the maximum value of a linear function cannot occur at a point inside the feasible region. The maximum value will always occur at one of the vertices.

Q: How do I determine which vertex gives the maximum value of a linear function?

A: To determine which vertex gives the maximum value of a linear function, you need to evaluate the function at each of the vertices and compare the resulting values. The vertex that gives the highest value is the one that corresponds to the maximum value of the function.

Q: Can I use a calculator to find the maximum value of a linear function?

A: Yes, you can use a calculator to find the maximum value of a linear function. Simply substitute the coordinates of each vertex into the function and calculate the resulting value.

Q: What is the importance of finding the maximum value of a linear function in a feasible region?

A: Finding the maximum value of a linear function in a feasible region is important because it helps to determine the optimal solution to a problem. In many real-world problems, the goal is to maximize a certain quantity, and finding the maximum value of a linear function is a key step in achieving this goal.

Conclusion

In conclusion, finding the maximum value of a linear function in a feasible region is an important step in solving many real-world problems. By understanding the concept of a feasible region and the significance of the vertices, you can use the techniques outlined in this article to find the maximum value of a linear function.

Frequently Asked Questions

  • What is a feasible region?
  • What are the vertices of a feasible region?
  • How do I find the maximum value of a linear function in a feasible region?
  • What is the significance of the vertices in finding the maximum value of a linear function?
  • Can the maximum value of a linear function occur at a point inside the feasible region?
  • How do I determine which vertex gives the maximum value of a linear function?
  • Can I use a calculator to find the maximum value of a linear function?
  • What is the importance of finding the maximum value of a linear function in a feasible region?

Answer Key

  • A feasible region is the set of all possible solutions to a system of linear inequalities.
  • The vertices of a feasible region are the points where the boundary lines intersect.
  • To find the maximum value of a linear function in a feasible region, you need to evaluate the function at each of the vertices of the feasible region.
  • The vertices are significant because they represent the extreme points of the feasible region.
  • No, the maximum value of a linear function cannot occur at a point inside the feasible region.
  • To determine which vertex gives the maximum value of a linear function, you need to evaluate the function at each of the vertices and compare the resulting values.
  • Yes, you can use a calculator to find the maximum value of a linear function.
  • Finding the maximum value of a linear function in a feasible region is important because it helps to determine the optimal solution to a problem.