A farmer wants to fence a rectangular area of 200 m^2. What dimensions will mini

A farmer wants to fence a rectangular area of 200 m^2. What dimensions will minimize the fencing required? (2021)

A farmer wants to fence a rectangular area of 200 m^2. What dimensions will minimize the fencing required? (2021)

Step 1: Understand that the farmer wants to create a rectangular area that is 200 square meters.
Step 2: Recall that the area of a rectangle is calculated by multiplying its length (L) by its width (W). So, L * W = 200.
Step 3: To minimize the amount of fencing (perimeter), we need to find the dimensions that give the smallest perimeter for the fixed area.
Step 4: The formula for the perimeter (P) of a rectangle is P = 2L + 2W.
Step 5: To minimize the perimeter while keeping the area constant, we can use the fact that a square has the smallest perimeter for a given area.
Step 6: Since the area is 200 m^2, we can find the side length of a square with this area by taking the square root of 200, which is approximately 14.14 m.
Step 7: However, since we are looking for a rectangle, we can set L = W to find the dimensions that minimize the perimeter.
Step 8: The dimensions that minimize the fencing required are approximately 14.14 m by 14.14 m.

Optimization of Area and Perimeter – The problem tests the understanding of how to minimize the perimeter of a rectangle while maintaining a fixed area, specifically recognizing that a square has the smallest perimeter for a given area.
Geometric Properties – It assesses knowledge of the properties of rectangles and squares, particularly that for a fixed area, a square minimizes the perimeter.