Final answer:
The farmer with 250 feet of fencing to create a rectangular pen would set up an equation 125 = L + W, representing the sum of the length and width, each multiplied by 2. Infinite solutions exist, but to maximize the area within the given perimeter, the pen should be a square with each side measuring 62.5 feet.
Step-by-step explanation:
The question involves a farmer who has 250 feet of fencing to create a rectangular pen. To solve the problem, we need to use algebra and an understanding of the properties of rectangles. Let's label the lengths of the two pairs of parallel sides of the rectangle as 'L' for the longer sides and 'W' for the shorter sides. As the perimeter P of a rectangle is twice the sum of its length and width (P = 2L + 2W), and the farmer has 250 feet of fencing (P = 250), our equation becomes 250 = 2L + 2W. Dividing each term by 2 simplifies the equation to 125 = L + W.
This is a linear equation with two variables. Without additional constraints, there are infinite solutions that will satisfy this equation because the farmer could choose various combinations of the length and width that add up to 125. However, the farmer may want to maximize the area of the rectangle, which involves calculus or another method to find the maximum area given a fixed perimeter. If maximizing the area, the farmer will find that a square is the most efficient shape, meaning L = W, and thus both will equal 62.5 feet.