Understanding the Square Park and Roads Problem

This problem involves calculating the area of two paths (roads) running through the center of a square park and then finding the cost to gravel these paths. The park is square, meaning its length and breadth are equal. The roads run parallel to the sides of the park, intersecting in the middle.

Park and Road Dimensions Analysis

Let's break down the given dimensions:

• Side of the square park = 20 meters (m).
• Width of each road = 2 meters (m).
• There are two roads, one running parallel to the length and the other parallel to the breadth, both in the middle.

When two paths run through the middle of a rectangle or square, parallel to the sides, they will overlap in the center. The area of this overlap is the area of a square formed by the intersection of the two roads.

Calculating the Area of the Paths

To find the total area of the paths, we can calculate the area of each path separately and then subtract the area of the central overlapping part, as it is counted twice.

The area of the road running parallel to the length is its length multiplied by its width. The length of this road within the park is equal to the side of the park.

• Length of the first road = Side of the park = 20 m
• Width of the first road = 2 m
• Area of the first road = Length × Width = \(20 \text{ m} \times 2 \text{ m} = 40 \text{ m}^2\).

Similarly, the area of the road running parallel to the breadth is its length (equal to the side of the park) multiplied by its width.

• Length of the second road = Side of the park = 20 m
• Width of the second road = 2 m
• Area of the second road = Length × Width = \(20 \text{ m} \times 2 \text{ m} = 40 \text{ m}^2\).

The two roads intersect in the middle. The shape of the intersection is a square with sides equal to the width of the roads.

• Side of the intersection square = Width of the road = 2 m
• Area of the intersection = Side × Side = \(2 \text{ m} \times 2 \text{ m} = 4 \text{ m}^2\).

The total area of the paths that needs gravelling is the sum of the areas of the two roads minus the area of the intersection (because it's included in both road area calculations).

Total Area of Paths = (Area of first road) + (Area of second road) - (Area of intersection)

Total Area of Paths = \(40 \text{ m}^2 + 40 \text{ m}^2 - 4 \text{ m}^2\)

Total Area of Paths = \(80 \text{ m}^2 - 4 \text{ m}^2\)

Total Area of Paths = \(76 \text{ m}^2\)

Calculating the Cost of Gravelling the Paths

The cost of gravelling is given per square meter. We know the total area of the paths and the rate.

• Area of paths = 76 m²
• Rate of gravelling = Rs. 100 per m²

Cost of gravelling = Area of paths × Rate per m²

Cost = \(76 \text{ m}^2 \times \text{Rs. } 100/\text{m}^2\)

Cost = Rs. 7600

So, the total cost of gravelling the paths is Rs. 7,600.

Revision Table: Square Park with Intersecting Roads

Additional Information: Areas of Paths in Rectangles/Squares

Calculating areas of paths within a rectangle or square is a common geometry problem. Here are some key points:

• Paths along the boundary: If a path runs just inside or outside the boundary of a rectangle/square, the area is calculated by finding the area of the larger rectangle/square (including the path) and subtracting the area of the smaller rectangle/square (the inner area).
• Paths intersecting in the middle: As seen in this problem, if two paths run parallel to the sides and intersect in the middle, the area of the intersection needs to be subtracted once from the sum of the individual path areas because it is counted in both.
• Units: Always ensure that all dimensions are in the same units before performing calculations. The area will be in square units (e.g., m²), and the cost will be the product of the area and the rate per square unit.
• Visualisation: Drawing a diagram of the park and the roads can be very helpful in understanding which areas overlap and need to be adjusted for.