Solving for the Path Width Surrounding a Circular Pond
When faced with the task of determining the width of a path surrounding a circular pond, we can follow a systematic mathematical approach to find the solution. This article will guide you through the process step by step, providing clear explanations and formulas that are easy to follow. Whether you are an environmental planner or just curious about geometric problems, this method will help you understand the underlying principles.
Introduction to the Problem
The problem at hand is to find the width of the path that surrounds a circular pond. This requires us to calculate relevant areas and understand the relationship between the pond's radius and the path's dimensions. Here, we will discuss the formulas and calculations necessary to solve this problem accurately.
Calculating the Area of the Pond
The area of a circle is given by the formula:
Formula for Area of a Circle
A_{text{pond}} pi r^2
where r is the radius of the pond.
Defining the Radius of the Entire Circle including the Path
Let the width of the path be w. Thus, the radius of the entire circle, including the pond and the path, is R r w.
Calculating the Area of the Larger Circle
The area of the larger circle, which includes both the pond and the path, can be calculated using the formula for the area of a circle:
Formula for Area of the Larger Circle
A_{text{total}} pi R^2 pi (r w)^2
Calculating the Area of the Path
The area of the path is the difference between the area of the larger circle and the area of the pond:
Formula for Area of the Path
A_{text{path}} A_{text{total}} - A_{text{pond}} pi (r w)^2 - pi r^2
Simplifying this, we get:
Simplified Formula for the Path Area
A_{text{path}} pi (r^2 2rw w^2) - pi r^2 pi (2rw w^2)
Setting the area of the path equal to the area of the pond:
Equate the Areas
pi (2rw w^2) pi r^2
Dividing both sides by pi:
Simplify the Equation
2rw w^2 r^2
Rearranging the equation:
Rearranged Equation
w^2 2rw - r^2 0
Solving the Quadratic Equation
We can use the quadratic formula to solve for w:
Quadratic Formula
w frac{-2r pm sqrt{(2r)^2 - 4 cdot 1 cdot (-r^2)}}{2 cdot 1}
Simplifying this, we get:
Simplified Quadratic Solution
w frac{-2r pm sqrt{4r^2 4r^2}}{2} frac{-2r pm 2rsqrt{2}}{2} -r pm rsqrt{2}
Since the width cannot be negative, we have:
Final Solution
w r(sqrt{2} - 1)
Therefore, the width of the path is given by:
Final Width Calculation
w r(sqrt{2} - 1) approx 0.414r
Conclusion
The width of the path surrounding a circular pond can be succinctly determined by the formula w r(sqrt{2} - 1). This result provides a practical solution to the problem of calculating the path width in terms of the pond's radius.
Understanding these steps can be invaluable for solving similar geometric problems. Whether you are designing a park or simply interested in the mathematical principles, this method offers clarity and precision.