Solving for the Width of a Uniformly Widen Path: A Comprehensive Guide
Understanding how to calculate the width of a path that surrounds a rectangular field, especially when the width is uniform, is crucial for various applications, such as landscaping, construction, and urban planning. In this article, we’ll explore the method to find the width of a path around a rectangular field, focusing on both uniform and non-uniform path widths. We’ll use two examples to illustrate the process.
Example 1: Uniform Width Path
A rectangular field is 16 meters long and 10 meters wide. There is a path of uniform width all around it, and the area of the path is 120 square meters. We need to find the width of this path.
Step-by-Step Solution:
1. Calculate the Area of the Rectangular Field
The area of the given rectangular field is calculated as:
pArea of the field length times; width 16 m times; 10 m 160 m2/p
2. Define the Variables:
Let the width of the path be (x) meters. The dimensions of the outer rectangle including the path will then be:
Length: (16 2x) meters
Width: (10 2x) meters
3. Calculate the Area of the Outer Rectangle
pArea of the outer rectangle (16 2x)(10 2x) 160 32x 2 4x2 160 52x 4x2/p
4. Set Up the Equation for the Area of the Path
The area of the path is given as 120 square meters. The area of the path can be expressed as:
pArea of the path Area of the outer rectangle - Area of the field/p
Therefore, substituting the known values:
p160 52x 4x2 - 160 120/p
5. Simplify and Solve for (x)
p4x2 52x - 120 0/p
Dividing the entire equation by 4:
px2 13x - 30 0/p
Factoring the quadratic equation:
p(x 15)(x - 2) 0/p
This gives us:
px -15 not valid (since width cannot be negative)brx 2 valid (since width must be positive)/p
Conclusion:
pThe width of the path is 2 meters./p
Example 2: Non-Uniform Width Path
A rectangular field is 20 meters long and 15 meters wide. There is a path of uniform width all around it, and the area of the path is 450 square meters. However, the path is not uniform in width, as it has rounded corners (circular arcs) and the path’s width is constant only at the straight boundaries.
Step-by-Step Solution:
1. Define the Variables:
Let the width of the path be (w) meters. The area of the path can be represented using the formula (πw^2 - 70w 450).
2. Solve for (w)
Rearranging the formula:
pπw2 - 70w - 450 0/p
Solving this quadratic equation using the quadratic formula gives us:
pw frac{-b pm sqrt{b2 - 4ac}}{2a}/p
Where (a π), (b -70), and (c -450).
3. Calculate the Width of the Path
Substituting the values:
pw frac{-(-70) pm sqrt{(-70)2 - 4(π)(-450)}}{2(π)}br frac{70 pm sqrt{4900 1800π}}{2π}br frac{70 pm sqrt{4900 5654.87}}{2π}br frac{70 pm sqrt{10554.87}}{2π}br frac{70 pm 102.74}{6.28}br frac{172.74}{6.28} or frac{-32.74}{6.28}br 27.52 or -5.21 (not valid)/p
The valid solution for the width of the path is approximately (5.21) meters.
Conclusion
The width of a path around a rectangular field, whether uniform or non-uniform, can be calculated using different methods depending on the specific conditions. The examples provided illustrate the process for both scenarios. Understanding these methods is essential for solving practical problems related to landscaping and construction. If you have any further questions or need additional assistance, feel free to reach out.