Final answer:
To minimize fencing for a rectangular field with an area of 200,000 m² divided by a fence into two halves, the field should be shaped with dimensions 447.21 m by 894.42 m.
Step-by-step explanation:
The subject matter here is Mathematics, specifically dealing with the optimization of areas and perimeters within geometric shapes. The student's question revolves around a real-world application of these principles, where they are seeking to minimize the fencing required for a rectangular field of a given area. The farmer's goal is to find the dimensions of the field that will achieve this with the added constraint of dividing the field in half with an additional fence.
To solve this, we must recognize that for a fixed area, the rectangle with the lowest perimeter is a square. Since the area is 200,000 m², the side length of the square would be √200,000 m, which is approximately 447.21 m. However, since the farmer wants to divide the field in half with a fence parallel to one of the sides, the resulting shape will be two adjacent squares, each with a side length of 447.21 m. Therefore, the dimensions that minimize the fencing required for a rectangular field would be 447.21 m x 894.42 m, with an additional fence of 447.21 m placed in the middle to divide the field into two equal halves.