Solving the Square Garden Puzzle with Walkways: A Comprehensive Guide
When tackling mathematical problems related to areas and dimensions, it's important to break down the problem into manageable steps. In this article, we'll explore a complex scenario involving a square garden with walkways. Specifically, we'll determine the dimensions of the planted area, given the total area of the garden including the walkways and the width of the walkway.
The Problem Statement
The problem involves a square garden with a walkway 5 feet wide around its corner edges. The total area of the garden including the walkways is 10,000 square feet. Our goal is to determine the dimensions of the planted area.
Step-by-Step Solution
Let's begin by breaking down the information given and solving the problem step-by-step.
Step 1: Determine the Side Length of the Total Garden Area
The total area of the garden including the walkways is given as 10,000 square feet. Since the garden is square-shaped, we can find the side length by taking the square root of the total area.
Mathematically, this can be expressed as:
Side length of the total garden area √10,000 100 feet
Step 2: Analyze the Walkway Layout
The walkway is described as being 5 feet wide around the corner edges. This means the walkway surrounds the garden but not entirely. Specifically, it is described as being 5 feet wide around the corner edges. This is key to understanding the layout of the garden and the walkway.
Step 3: Determine the Planted Area Dimensions
To find the dimensions of the planted area, we need to account for the walkway. The walkway adds 5 feet to the dimensions on both sides where it meets the corner. Therefore, the dimensions of the planted area will be reduced by 10 feet (5 feet from each side) compared to the total garden area.
Mathematically, this can be expressed as:
Planted area side length Total garden side length - 10 feet
Planted area side length 100 feet - 10 feet 90 feet
Step 4: Calculate the Area of the Planted Area
Now that we have the side length of the planted area, we can calculate its area. Since the planted area is also a square, we can find its area by squaring the side length.
Area of the planted area 90 feet × 90 feet 8,100 square feet
Step 5: Conclusion
Therefore, the dimensions of the planted area are 90 feet by 90 feet, and the area of the planted area is 8,100 square feet.
Frequently Asked Questions
Q: Is the walkway all around the garden?
No, the walkway is described as being around the corner edges of the garden, not entirely surrounding it. This is a critical detail to understand the layout correctly.
Q: How do I set up the sketch for the garden with walkway?
One way to set up the sketch is to draw a larger square for the total garden area and a smaller square within it for the planted area. The 5 feet wide walkway can be represented as an overlapping layer around the corner edges of the smaller square.
Q: Can you provide additional hints for solving similar problems?
When solving similar problems, it's crucial to carefully read the problem statement. Identify the key elements such as the total area, the dimensions of the walkway, and the layout. Break down the problem into manageable steps and always consider the implications of the layout on the dimensions.
Conclusion
In conclusion, understanding the layout and dimensions of a garden with walkways involves careful calculation and visualization. By breaking down the problem and following the steps outlined above, you can accurately determine the dimensions and area of the planted area. This type of problem-solving is not only useful in real-world scenarios but also in advancing your mathematical skills.