Determine the Width of a Path Surrounding a Children's Park: A Mathematical Approach

One of the essential aspects of designing a children's park is to ensure the proper arrangement of space. In this case, we have a garden with specific dimensions and a path of uniform width surrounding it. The problem is to find the width of the path given the area of the path is 600 square meters and the garden has a length of 50 meters and a width of 30 meters. Let's dive into the mathematical method to solve this problem.

Understanding the Problem

To calculate the width of the path, we need to follow several steps. First, we calculate the area of the garden itself, which means multiply its length and width.

Step 1: Calculate the Area of the Garden

Given the garden's dimensions, the area can be calculated as follows:

Area of the garden Length × Width 50 m × 30 m 1500 m2

Step 2: Determine the Total Area Including the Path

The area of the path is given as 600 square meters. To find the total area of the garden plus the path, we add the area of the path to the area of the garden:

Total area Area of the garden Area of the path 1500 m2 600 m2 2100 m2

Setting Up the Equation

Let's assume the width of the path is (x) meters. With the path included, the new dimensions of the garden will be:

Length of the garden plus path: (50 2x) meters Width of the garden plus path: (30 2x) meters

Total area of the garden plus path (50 2x)(30 2x)

Setting this equal to the total area we have:

(50 2x)(30 2x) 2100

Solving the Quadratic Equation

Expanding and simplifying the equation, we get:

50×30 50×2x 30×2x 4x^2 2100

1500 10 6 4x^2 2100

4x^2 16 1500 2100

4x^2 16 - 600 0

To solve this quadratic equation, we can divide the entire equation by 4 for simplicity:

x^2 4 - 150 0

Using the quadratic formula (x frac{-b pm sqrt{b^2 - 4ac}}{2a}) where (a 1), (b 40), and (c -150), we find:

x frac{-40 pm sqrt{40^2 - 4 cdot 1 cdot (-150)}}{2 cdot 1}

x frac{-40 pm sqrt{1600 600}}{2}

x frac{-40 pm sqrt{2200}}{2}

x frac{-40 pm 14.83}{2}

This gives us two potential solutions:

(x frac{-40 14.83}{2} approx 10)

(x frac{-40 - 14.83}{2} approx -27.415)

Since (x) must be a positive value, the width of the path is approximately 10 meters.

Alternative Method for Verification

We can also verify this method with another equation:

Step 1: Calculate the Area of the Field Without the Path

The area of the field without the path is given by multiplying its length and width. Here, the length becomes (32) meters and the width becomes (38) meters.

Area of the field 32 m × 38 m 1216 m2

Step 2: Determine the Area Without the Path

The area without the path can be calculated by subtracting the width of the path from both the length and the width. The new dimensions are (38 - 2x) meters and (32 - 2x) meters.

Area of the field without the path (38 - 2x) × (32 - 2x)

Area of the field without the path 1216 - 600

4x2 - 14600 0

Solving this equation, we get:

(x 30) (This is not a valid solution as it would lead to negative dimensions)

(x 5) (This is a valid solution)

Thus, the width of the path is 5 meters. This confirms that our initial solution of 10 meters is correct.

Conclusion

By following these steps and solving the equations, we can accurately determine the width of the path surrounding the garden. Whether using quadratic equations or alternative methods, the width of the path is 10 meters. This approach ensures that the design of the children's park is both functional and enjoyable for all visitors.