Jones Needs To Build A Rectangular Fence In His Backyard To Keep His Dog Safe. The Length Of The Fence { ( Y )$}$ Should Be At Least 60 Ft, And The Perimeter Should Be No More Than 260 Ft. Which System Of Inequalities And Graph Represent The

Introduction

Jones is a dog owner who wants to build a rectangular fence in his backyard to keep his furry friend safe. However, he has some constraints to consider. The length of the fence should be at least 60 ft, and the perimeter should not exceed 260 ft. In this article, we will explore the system of inequalities and graph that represent Jones' rectangular fence problem.

The Problem

Jones wants to build a rectangular fence with a length of at least 60 ft and a perimeter of no more than 260 ft. Let's denote the length of the fence as ${y}$ and the width as ${x}$. The perimeter of a rectangle is given by the formula ${P = 2x + 2y}$. Since the perimeter should not exceed 260 ft, we can write the inequality:

${2x + 2y \leq 260}$

We can simplify this inequality by dividing both sides by 2:

${x + y \leq 130}$

The Length Constraint

The length of the fence should be at least 60 ft, so we can write the inequality:

${y \geq 60}$

The System of Inequalities

Now, we have two inequalities that represent Jones' rectangular fence problem:

${x + y \leq 130}$ ${y \geq 60}$

Graphing the Inequalities

To graph the inequalities, we can use a coordinate plane with ${x}$ and ${y}$ axes. We will graph the lines ${x + y = 130}$ and ${y = 60}$ and then shade the regions that satisfy the inequalities.

Graphing the Line ${x + y = 130}$

The line ${x + y = 130}$ can be graphed by finding two points that satisfy the equation. Let's find the points where ${x = 0}$ and ${y = 0}$.

When ${x = 0}$, we have:

${0 + y = 130}$ ${y = 130}$

When ${y = 0}$, we have:

${x + 0 = 130}$ ${x = 130}$

So, the points ${(0, 130)}$ and ${(130, 0)}$ satisfy the equation. We can plot these points on the coordinate plane and draw a line through them.

Graphing the Line ${y = 60}$

The line ${y = 60}$ is a horizontal line that passes through the point ${(0, 60)}$. We can plot this point on the coordinate plane and draw a line through it.

Shading the Regions

To shade the regions that satisfy the inequalities, we need to determine which regions are above or below the lines. Let's analyze the inequalities:

${x + y \leq 130}$ ${y \geq 60}$

The first inequality is a linear inequality, and the second inequality is a linear inequality with a horizontal line. We can shade the region below the line ${x + y = 130}$ and above the line ${y = 60}$.

The Graph

The graph of the system of inequalities is a shaded region that represents the possible values of ${x}$ and ${y}$ that satisfy the constraints. The region is bounded by the lines ${x + y = 130}$ and ${y = 60}$.

Conclusion

In this article, we explored the system of inequalities and graph that represent Jones' rectangular fence problem. We wrote the inequalities that represent the length and perimeter constraints and graphed the lines and shaded the regions that satisfy the inequalities. The graph represents the possible values of ${x}$ and ${y}$ that satisfy the constraints, and it can be used to help Jones choose the dimensions of his rectangular fence.

Final Answer

The final answer is the graph of the system of inequalities, which represents the possible values of ${x}$ and ${y}$ that satisfy the constraints.

References

• [1] "Linear Inequalities" by Math Open Reference
• [2] "Graphing Linear Inequalities" by Khan Academy
• [3] "Rectangular Fences" by Wolfram Alpha

Additional Resources

• [1] "Mathematics for Elementary Teachers" by John F. Harper
• [2] "Graphing Linear Equations and Inequalities" by IXL
• [3] "Rectangular Fences" by Mathway
  

Introduction

In our previous article, we explored the system of inequalities and graph that represent Jones' rectangular fence problem. Jones wants to build a rectangular fence with a length of at least 60 ft and a perimeter of no more than 260 ft. In this article, we will answer some frequently asked questions about the problem.

Q&A

Q: What is the length constraint for the fence?

A: The length of the fence should be at least 60 ft.

Q: What is the perimeter constraint for the fence?

A: The perimeter of the fence should not exceed 260 ft.

Q: How do we represent the length and perimeter constraints mathematically?

A: We can represent the length constraint as the inequality ${y \geq 60}$, where ${y}$ is the length of the fence. We can represent the perimeter constraint as the inequality ${x + y \leq 130}$, where ${x}$ is the width of the fence.

Q: How do we graph the inequalities?

A: We can graph the line ${x + y = 130}$ and the line ${y = 60}$, and then shade the regions that satisfy the inequalities.

Q: What is the graph of the system of inequalities?

A: The graph of the system of inequalities is a shaded region that represents the possible values of ${x}$ and ${y}$ that satisfy the constraints.

Q: How can we use the graph to help Jones choose the dimensions of his rectangular fence?

A: We can use the graph to find the possible values of ${x}$ and ${y}$ that satisfy the constraints. We can then choose the dimensions of the fence that are within the shaded region.

Q: What are some real-world applications of this problem?

A: This problem has many real-world applications, such as designing a rectangular garden, a swimming pool, or a building.

Q: How can we extend this problem to include more constraints?

A: We can extend this problem by adding more constraints, such as a minimum or maximum width, a minimum or maximum height, or a specific shape for the fence.

Additional Questions and Answers

Q: What is the difference between a linear inequality and a linear equation?

A: A linear inequality is an inequality that can be written in the form ${ax + by \leq c}$ or ${ax + by \geq c}$, where ${a}$, ${b}$, and ${c}$ are constants. A linear equation is an equation that can be written in the form ${ax + by = c}$, where ${a}$, ${b}$, and ${c}$ are constants.

Q: How do we solve a system of linear inequalities?

A: We can solve a system of linear inequalities by graphing the lines and shading the regions that satisfy the inequalities.

Q: What is the importance of graphing linear inequalities?

A: Graphing linear inequalities is important because it helps us visualize the solution set and make decisions based on the constraints.

Conclusion

In this article, we answered some frequently asked questions about Jones' rectangular fence problem. We discussed the length and perimeter constraints, how to represent them mathematically, and how to graph the inequalities. We also discussed some real-world applications of the problem and how to extend it to include more constraints.

Final Answer

The final answer is the graph of the system of inequalities, which represents the possible values of ${x}$ and ${y}$ that satisfy the constraints.

References

• [1] "Linear Inequalities" by Math Open Reference
• [2] "Graphing Linear Inequalities" by Khan Academy
• [3] "Rectangular Fences" by Wolfram Alpha

Additional Resources

• [1] "Mathematics for Elementary Teachers" by John F. Harper
• [2] "Graphing Linear Equations and Inequalities" by IXL
• [3] "Rectangular Fences" by Mathway