Determine the Width of a Path in a Children's Park: A Geometric Problem

Imagine a charming children's park that features a beautifully designed garden measuring 50 meters in width and 30 meters in length. Around this garden, a path of uniform width is created. To understand the design's geometric intricacies—specifically the width of this path—let's explore a real-world problem involving area calculations.

The Problem Setup

Suppose the total area of the path around the garden is 600 square meters. If the width of the path is to be determined, we can establish the following equation and solve for the unknown width.

The Equation Derivation

Let the width of the path be x meters. We can break down the problem into a few key steps:

The total area of the field including the path is the sum of the area of the field itself and the area of the path. The area of the field excluding the path is a smaller rectangle inside the larger one, with dimensions reduced by twice the width of the path on each side.

With these considerations in mind, we can establish the following:

Step-by-Step Solution

Calculate the total area of the field: Given dimensions of the garden:Length: 38 metersWidth: 32 metersTherefore, the area of the field is 38 x 32 1216 square meters. Calculate the area of the garden excluding the path: The dimensions of the garden excluding the path are reduced by twice the width of the path on each side. Thus, the new dimensions are: Length excluding the path: 38 - 2x Width excluding the path: 32 - 2x The area of the field excluding the path is (38 - 2x) x (32 - 2x) 1216 - 144x^2 Calculate the area of the path: The area of the path is the difference between the total area of the field and the area excluding the path: 1216 - (1216 - 144x^2) 144x^2 600 Solve for x: The algebraic steps to find x are as follows: 144x^2 600 x^2 600 / 144 25 / 9 x 5 / 3 (approximately) The exact value of x is 5 meters. We can discard x 30 meters as it would make the dimensions of the field inside the path negative, which is not possible.

Conclusion

The width of the path in this children's park is 5 meters. This solution adheres to the geometric constraints and practical considerations of the problem, providing a clear and accurate answer.

Relevant Formulas and Equations

To summarize, the relevant formulas used in this problem are:

Total area of the field 38 x 32 1216 square meters

Area of the path 144x^2 600 square meters

Equation for x: x^2 - 35x 150 0

Solution: x 5 m

This problem effectively uses the principles of area calculation in geometry, making it a valuable exercise for students and professionals alike.