Answer:

To divide the land into two separate areas using 300 meters of fencing material, ensuring both areas have the same area but different perimeters, we can create two shapes with equal areas but different perimeters. One possible solution is to create a rectangle and a circle.

Here's how we can divide the land:

1. Rectangle:

Let's allocate one side of the land for the rectangle. Since we have 300 meters of fencing material, we can use 100 meters for the length of the rectangle. This leaves us with 200 meters of fencing material.

To find the width of the rectangle, we can use the formula for the perimeter of a rectangle:

Perimeter = 2 * (length + width)

Since the perimeter of the rectangle should be different from the perimeter of the circle, we can choose a different width for the rectangle. Let's say we choose a width of 50 meters.

The perimeter of the rectangle would be:

Perimeter = 2 * (100 + 50) = 300 meters

The area of the rectangle is given by:

Area = length * width = 100 * 50 = 5000 square meters

2. Circle:

With the remaining 200 meters of fencing material, we can create a circle. The circumference of a circle is given by the formula:

Circumference = 2 * π * radius

To find the radius, we can rearrange the formula:

radius = Circumference / (2 * π)

Since we want the circle to have the same area as the rectangle, we can calculate the area of the rectangle (5000 square meters) and use it to find the radius of the circle.

Area of the circle = π * radius^2

5000 = π * radius^2

Solving for radius:

radius^2 = 5000 / π

radius ≈ √(5000 / π) ≈ 39.86 meters

The circumference of the circle would be:

Circumference = 2 * π * radius ≈ 2 * 3.14 * 39.86 ≈ 250.96 meters

Therefore, by dividing the land into a rectangle with a perimeter of 300 meters and an area of 5000 square meters, and a circle with a circumference of approximately 250.96 meters and the same area of 5000 square meters, we can achieve two separate areas with equal areas but different perimeters.