Understanding the Original Garden

Oct 13, 2025 by miodeya.com 145 views

Introduction

Miles has a square garden in his backyard, which he decides to modify by decreasing the size of the garden by 1 foot on each side to create a gravel border. This simple yet intriguing scenario presents an opportunity to delve into the world of mathematics, specifically focusing on the properties of squares and the impact of changes in dimensions on their areas. In this article, we will explore the mathematical concepts underlying Miles' garden transformation and derive an equation to represent the new area of the garden.

Understanding the Original Garden

Let's assume that the original side length of Miles' square garden is x feet. Since the garden is a square, its area can be calculated using the formula:

Area = side length²

In this case, the area of the original garden is:

Area = x²

The Gravel Border: A Reduction in Size

Miles decides to decrease the size of the garden by 1 foot on each side to create a gravel border. This means that the new side length of the garden will be (x - 1) feet. To find the area of the new garden, we can use the same formula:

Area = side length²

Substituting the new side length, we get:

Area = (x - 1)²

The New Area of the Garden

We are given that the area of the new garden is 25 feet². We can set up an equation using this information:

(x - 1)² = 25

Solving the Equation

To solve for x, we can start by taking the square root of both sides of the equation:

√(x - 1)² = √25

This simplifies to:

x - 1 = ±5

Since the side length of the garden cannot be negative, we take the positive value:

x - 1 = 5

Adding 1 to both sides, we get:

x = 6

The Original Side Length

Now that we have found the value of x, we can determine the original side length of Miles' square garden. Since x = 6, the original side length was 6 feet.

The Area of the Original Garden

Using the formula for the area of a square, we can calculate the area of the original garden:

Area = x² = 6² = 36

Conclusion

In this article, we explored the mathematical concepts underlying Miles' garden transformation. We started with the original garden, which had a side length of x feet and an area of x². After Miles decreased the size of the garden by 1 foot on each side, the new side length became (x - 1) feet, and the area became (x - 1)². We were given that the area of the new garden was 25 feet², and we used this information to derive an equation and solve for x. The original side length of Miles' square garden was found to be 6 feet, and the area of the original garden was calculated to be 36 square feet.

Further Exploration

This problem can be extended in various ways, such as:

What if the garden is not a square? How would the area change if the garden were a rectangle or a triangle?
What if the gravel border is not 1 foot wide? How would the area change if the gravel border were 2 feet or 3 feet wide?
What if the garden is not in Miles' backyard? How would the area change if the garden were in a different location, such as a park or a farm?

These questions and others like them can lead to further exploration of mathematical concepts and their applications in real-world scenarios.

Real-World Applications

The concepts explored in this article have real-world applications in various fields, such as:

Architecture: Understanding the properties of squares and rectangles is crucial in designing buildings and other structures.
Engineering: Calculating areas and volumes is essential in designing and building infrastructure, such as roads, bridges, and buildings.
Landscaping: Understanding the area of a garden or a lawn is important in designing and maintaining outdoor spaces.

By applying mathematical concepts to real-world scenarios, we can gain a deeper understanding of the world around us and make informed decisions in various fields.

Conclusion

In conclusion, Miles' square garden presents a simple yet intriguing scenario that allows us to explore mathematical concepts and their applications in real-world scenarios. By understanding the properties of squares and rectangles, we can calculate areas and volumes, which is essential in various fields, such as architecture, engineering, and landscaping. The concepts explored in this article can be extended and applied to various scenarios, leading to further exploration and understanding of mathematical concepts.

Introduction

In our previous article, we explored the mathematical concepts underlying Miles' garden transformation. We started with the original garden, which had a side length of x feet and an area of x². After Miles decreased the size of the garden by 1 foot on each side, the new side length became (x - 1) feet, and the area became (x - 1)². We were given that the area of the new garden was 25 feet², and we used this information to derive an equation and solve for x. In this article, we will answer some of the most frequently asked questions related to Miles' square garden.

Q&A

Q: What is the original side length of Miles' square garden?

A: The original side length of Miles' square garden is 6 feet.

Q: What is the area of the original garden?

A: The area of the original garden is 36 square feet.

Q: How did Miles decrease the size of the garden?

A: Miles decreased the size of the garden by 1 foot on each side to create a gravel border.

Q: What is the new side length of the garden?

A: The new side length of the garden is (x - 1) feet, which is 5 feet.

Q: What is the area of the new garden?

A: The area of the new garden is 25 square feet.

Q: How did you derive the equation to represent the new area of the garden?

A: We used the formula for the area of a square, which is side length², and substituted the new side length, (x - 1), to get the equation (x - 1)² = 25.

Q: How did you solve for x?

A: We took the square root of both sides of the equation, √(x - 1)² = √25, and simplified to get x - 1 = ±5. Since the side length of the garden cannot be negative, we took the positive value, x - 1 = 5, and added 1 to both sides to get x = 6.

Q: What are some real-world applications of the concepts explored in this article?

A: The concepts explored in this article have real-world applications in various fields, such as architecture, engineering, and landscaping. Understanding the properties of squares and rectangles is crucial in designing buildings and other structures, calculating areas and volumes is essential in designing and building infrastructure, and understanding the area of a garden or a lawn is important in designing and maintaining outdoor spaces.

Q: Can you extend this problem to other shapes, such as rectangles or triangles?

A: Yes, we can extend this problem to other shapes, such as rectangles or triangles. For example, if the garden were a rectangle with a length of l feet and a width of w feet, the area would be lw. If the garden were a triangle with a base of b feet and a height of h feet, the area would be (1/2)bh.

Conclusion

In this article, we answered some of the most frequently asked questions related to Miles' square garden. We explored the mathematical concepts underlying the garden transformation and derived an equation to represent the new area of the garden. We also discussed some real-world applications of the concepts explored in this article and extended the problem to other shapes, such as rectangles and triangles.