Introduction

In this article, we will explore how to find the width of a circular path surrounding a circular flower bed, given the diameter of the flower bed and the area of the path. This is a common problem in geometry and landscaping, and understanding the mathematical principles behind it can be very useful for both students and professionals in fields such as gardening, architecture, and urban planning.

Problem Statement

Suppose you have a circular flower bed with a diameter of 22 meters. The flower bed is surrounded by a circular path of uniform width. If the area of the path is 80 square meters, how can you find the width of the path?

Solution

To solve this problem, we will break it down into several steps, using the principles of geometry and algebra.

Step 1: Calculate the Radius of the Flower Bed

The first step is to find the radius of the flower bed. Given that the diameter is 22 meters, we can calculate the radius as follows:

Radius of the flower bed: r 22 / 2 11 meters

Step 2: Define the Radius of the Outer Circle (Flower Bed Path)

Let W be the width of the path. The radius of the outer circle, which includes the flower bed and the path, is:

Radius of the outer circle: R r W 11 W

Step 3: Calculate the Area of the Flower Bed

The area of the flower bed is given by the formula for the area of a circle:

Area of the flower bed: A_{flower bed} π r^2 π (11^2) 121π square meters

Step 4: Calculate the Area of the Outer Circle (Flower Bed Path)

The area of the outer circle is:

Area of the outer circle: A_{outer} π R^2 π (11 W)^2

Step 5: Set Up the Equation for the Area of the Path

The area of the path is the area of the outer circle minus the area of the flower bed:

Area of the path: A_{path} A_{outer} - A_{flower bed} π (11 W)^2 - 121π

We know that the area of the path is 80 square meters, so we can set up the equation:

π (11 W)^2 - 121π 80

Step 6: Simplify the Equation

Dividing both sides of the equation by π (assuming π ≠ 0) and simplifying:

(11 W)^2 - 121 80 / π

(11 W)^2 121 80 / π

Using π ≈ 3.14, we find:

121 80 / 3.14 ≈ 121 25.48 ≈ 146.48

Thus:

(11 W)^2 146.48

Step 7: Solve for W

Take the square root of both sides:

11 W ≈ √146.48 ≈ 12.1

Subtract 11 from both sides to solve for W:

W ≈ 12.1 - 11 ≈ 1.1

Therefore, the width of the path is approximately 1.1 meters.

Conclusion

In this article, we have demonstrated how to find the width of a circular path surrounding a circular flower bed, given the diameter of the flower bed and the area of the path. By using the principles of geometry and algebra, we can accurately determine the width of the path, which is essential for designing and planning various landscaping projects.