Finding the Maximum Perimeter

Oct 13, 2025 by miodeya.com 106 views

Introduction

Mike wants to fence in part of his backyard, and he has 200 feet of fencing available. The length of the fenced-in area must be at least 20 feet long. In this article, we will explore the mathematical model that represents the possible perimeter of the yard.

The Inequality Model

The inequality that models the possible perimeter of the yard is given by:

2l + 2w \leq 200

where $l$ is the length of the fenced-in area and $w$ is the width. However, we are given that the length of the fenced-in area must be at least 20 feet long, so we can write:

l \geq 20

We can substitute this inequality into the first inequality to get:

2(20) + 2w \leq 200

Simplifying the inequality, we get:

40 + 2w \leq 200

Subtracting 40 from both sides, we get:

2w \leq 160

Dividing both sides by 2, we get:

w \leq 80

This means that the width of the fenced-in area must be less than or equal to 80 feet.

Graphing the Inequality

To visualize the possible values of $l$ and $w$ , we can graph the inequality on a coordinate plane. The inequality can be written as:

2l + 2w \leq 200

We can rewrite this inequality as:

l + w \leq 100

This is a linear inequality, and we can graph it on a coordinate plane. The graph of the inequality is a line with a slope of -1 and a y-intercept of 100.

Finding the Maximum Perimeter

To find the maximum perimeter of the yard, we need to find the maximum value of $l + w$ . Since the inequality is a linear inequality, the maximum value of $l + w$ will occur when $l$ and $w$ are at their maximum values.

From the inequality, we know that $l \geq 20$ and $w \leq 80$ . To maximize the perimeter, we want to maximize $l$ and minimize $w$ . Therefore, we can set $l = 20$ and $w = 80$ .

Substituting these values into the inequality, we get:

2(20) + 2(80) \leq 200

Simplifying the inequality, we get:

40 + 160 \leq 200

This is true, so the maximum perimeter of the yard is:

l + w = 20 + 80 = 100

Conclusion

In this article, we explored the mathematical model that represents the possible perimeter of Mike's backyard. We found that the inequality that models the possible perimeter of the yard is:

2l + 2w \leq 200

We also found that the width of the fenced-in area must be less than or equal to 80 feet. To maximize the perimeter, we want to maximize $l$ and minimize $w$ . Therefore, we can set $l = 20$ and $w = 80$ .

The maximum perimeter of the yard is 100 feet, which occurs when $l = 20$ and $w = 80$ .

References

[1] "Mathematics for the Nonmathematician" by Morris Kline
[2] "Calculus" by Michael Spivak

Glossary

Inequality: A statement that two expressions are not equal.
Linear inequality: An inequality that can be written in the form $ax + by \leq c$ .
Perimeter: The distance around a shape.
Width: The distance between two parallel sides of a shape.
Fencing the Backyard: A Mathematical Approach - Q&A =====================================================

Introduction

In our previous article, we explored the mathematical model that represents the possible perimeter of Mike's backyard. We found that the inequality that models the possible perimeter of the yard is:

2l + 2w \leq 200

where $l$ is the length of the fenced-in area and $w$ is the width. We also found that the width of the fenced-in area must be less than or equal to 80 feet.

In this article, we will answer some common questions that people may have about fencing the backyard.

Q: What is the maximum length of the fenced-in area?

A: The maximum length of the fenced-in area is 100 feet. This occurs when the width of the fenced-in area is 0 feet.

Q: What is the minimum width of the fenced-in area?

A: The minimum width of the fenced-in area is 0 feet. This occurs when the length of the fenced-in area is 100 feet.

Q: Can I fence in a rectangular area with a length of 20 feet and a width of 80 feet?

A: Yes, you can fence in a rectangular area with a length of 20 feet and a width of 80 feet. This is one of the possible solutions to the inequality.

Q: Can I fence in a square area with a side length of 50 feet?

A: No, you cannot fence in a square area with a side length of 50 feet. This is because the perimeter of the square would be 200 feet, which is greater than the available fencing.

Q: How can I maximize the area of the fenced-in area?

A: To maximize the area of the fenced-in area, you should make the length and width as large as possible. However, you should also make sure that the perimeter of the fenced-in area is less than or equal to 200 feet.

Q: Can I fence in a circular area?

A: No, you cannot fence in a circular area. This is because the perimeter of a circle is not a straight line, and it would be difficult to fence in a circular area with a fixed amount of fencing.

Q: How can I calculate the area of the fenced-in area?

A: To calculate the area of the fenced-in area, you can use the formula:

A = lw

where $A$ is the area, $l$ is the length, and $w$ is the width.

Conclusion

In this article, we answered some common questions that people may have about fencing the backyard. We found that the maximum length of the fenced-in area is 100 feet, and the minimum width is 0 feet. We also found that you can fence in a rectangular area with a length of 20 feet and a width of 80 feet, but you cannot fence in a square area with a side length of 50 feet.

References

[1] "Mathematics for the Nonmathematician" by Morris Kline
[2] "Calculus" by Michael Spivak

Glossary

Area: The amount of space inside a shape.
Length: The distance between two points on a shape.
Perimeter: The distance around a shape.
Width: The distance between two parallel sides of a shape.