Miles Has A Square Garden In His Backyard. He Decides To Decrease The Size Of The Garden By 1 Foot On Each Side In Order To Make A Gravel Border. After He Completes His Gravel Border, The Area Of The New Garden Is 25 Square Feet. In The Equation
Introduction
Miles has a square garden in his backyard, which he decides to decrease in size by 1 foot on each side to create a gravel border. This decision leads to a new area of the garden, which is 25 square feet. In this article, we will delve into the mathematical problem behind Miles' decision and explore the relationship between the original and new garden sizes.
The Original Garden Size
Let's assume that the original size of the garden is x feet by x feet, where x is a positive integer. Since the garden is square, the area of the original garden is given by the formula:
Area = x^2
We are not given the exact value of x, but we know that the new garden size is 25 square feet. This means that the new garden is a square with a side length of √25 = 5 feet.
The New Garden Size
Since Miles decreases the size of the garden by 1 foot on each side, the new garden size is (x-2) feet by (x-2) feet. The area of the new garden is given by the formula:
Area = (x-2)^2
We are given that the area of the new garden is 25 square feet, so we can set up the equation:
(x-2)^2 = 25
Solving the Equation
To solve the equation, we can start by taking the square root of both sides:
√(x-2)^2 = √25
This simplifies to:
x-2 = ±5
Since x is a positive integer, we can ignore the negative solution and focus on the positive solution:
x-2 = 5
Adding 2 to both sides gives us:
x = 7
The Original Garden Size
Now that we have found the value of x, we can find the original garden size. Since x = 7, the original garden size is 7 feet by 7 feet.
The Relationship Between the Original and New Garden Sizes
We can now explore the relationship between the original and new garden sizes. Since the new garden size is (x-2) feet by (x-2) feet, we can substitute x = 7 into this expression:
New garden size = (7-2) feet by (7-2) feet = 5 feet by 5 feet
This shows that the new garden size is a square with a side length of 5 feet, which is consistent with our earlier result.
Conclusion
In this article, we have explored the mathematical problem behind Miles' decision to decrease the size of his garden. We have found that the original garden size is 7 feet by 7 feet, and the new garden size is 5 feet by 5 feet. This shows that the relationship between the original and new garden sizes is a simple linear relationship, where the new garden size is 3 feet smaller than the original garden size on each side.
Further Exploration
This problem can be further explored by considering different values of x and analyzing the resulting relationships between the original and new garden sizes. Additionally, we can consider other shapes, such as rectangles or triangles, and explore the relationships between their original and new sizes.
Mathematical Concepts
This problem involves several mathematical concepts, including:
- Algebra: The problem involves solving a quadratic equation to find the value of x.
- Geometry: The problem involves understanding the properties of squares and rectangles, including their areas and perimeters.
- Problem-solving: The problem requires the ability to analyze a situation, identify the relevant mathematical concepts, and apply them to solve the problem.
Real-World Applications
This problem has several real-world applications, including:
- Landscaping: The problem can be applied to real-world situations where a gardener or landscaper needs to decrease the size of a garden or yard to create a border or other feature.
- Architecture: The problem can be applied to real-world situations where an architect needs to design a building or other structure with a specific size or shape.
- Engineering: The problem can be applied to real-world situations where an engineer needs to design a system or device with a specific size or shape.
Conclusion
In conclusion, Miles' square garden problem is a classic example of a mathematical problem that involves algebra, geometry, and problem-solving. The problem has several real-world applications and can be further explored by considering different values of x and analyzing the resulting relationships between the original and new garden sizes.
Introduction
In our previous article, we explored the mathematical problem behind Miles' decision to decrease the size of his garden. We found that the original garden size is 7 feet by 7 feet, and the new garden size is 5 feet by 5 feet. In this article, we will answer some of the most frequently asked questions about this problem.
Q: What is the relationship between the original and new garden sizes?
A: The relationship between the original and new garden sizes is a simple linear relationship, where the new garden size is 3 feet smaller than the original garden size on each side.
Q: How do I find the original garden size if I know the new garden size?
A: To find the original garden size, you can use the formula:
Original garden size = New garden size + 2
For example, if the new garden size is 5 feet by 5 feet, the original garden size would be:
Original garden size = 5 + 2 = 7 feet by 7 feet
Q: What if the new garden size is not a perfect square?
A: If the new garden size is not a perfect square, you can still find the original garden size by using the formula:
Original garden size = New garden size + 2
However, you will need to find the square root of the new garden size to get the side length.
Q: Can I apply this problem to other shapes, such as rectangles or triangles?
A: Yes, you can apply this problem to other shapes, such as rectangles or triangles. However, you will need to use different formulas to find the original size.
Q: What are some real-world applications of this problem?
A: Some real-world applications of this problem include:
- Landscaping: The problem can be applied to real-world situations where a gardener or landscaper needs to decrease the size of a garden or yard to create a border or other feature.
- Architecture: The problem can be applied to real-world situations where an architect needs to design a building or other structure with a specific size or shape.
- Engineering: The problem can be applied to real-world situations where an engineer needs to design a system or device with a specific size or shape.
Q: How can I practice solving this problem?
A: You can practice solving this problem by trying different values of x and analyzing the resulting relationships between the original and new garden sizes. You can also try applying the problem to other shapes, such as rectangles or triangles.
Q: What are some common mistakes to avoid when solving this problem?
A: Some common mistakes to avoid when solving this problem include:
- Not using the correct formula to find the original garden size
- Not considering the relationship between the original and new garden sizes
- Not checking for errors in the calculations
Conclusion
In conclusion, Miles' square garden problem is a classic example of a mathematical problem that involves algebra, geometry, and problem-solving. By understanding the relationship between the original and new garden sizes, you can apply this problem to real-world situations and practice solving it to improve your math skills.
Additional Resources
If you want to learn more about this problem or practice solving it, here are some additional resources:
- Online math tutorials and videos
- Math textbooks and workbooks
- Math apps and software
- Online math communities and forums
Final Thoughts
Miles' square garden problem is a fun and challenging math problem that can be applied to real-world situations. By understanding the relationship between the original and new garden sizes, you can improve your math skills and apply this problem to other areas of your life.