Final answer:
The farmer can maximize the enclosed area by building a square-shaped enclosure with each of the three fenced sides measuring 54.67 feet, resulting in a maximum area of approximately 2,989.45 square feet.
Step-by-step explanation:
The student has asked about the largest area that a farmer can enclose with a set length of fence when one side is already provided by an existing wall. This is a problem that requires knowledge of geometry and is specifically related to optimization problems, where we are maximizing the enclosed area. Given that the farmer has 164 feet of fencing and needs to fence only three sides of a rectangular area, the optimization occurs when the two sides perpendicular to the wall are equal in length, resulting in a square-shaped area for these three sides.
To maximize the area, the farmer should divide the available fencing into three equal parts (as two sides and one equal side parallel to the wall), which would be 164 feet / 3 = 54.67 feet for each side. However, since one side is already present (the wall), the farmer only needs to use the fencing for the remaining two sides and the side parallel to the wall. So the lengths of the two perpendicular sides will each be 54.67 feet, and the side parallel to the wall will also be 54.67 feet long. The largest area that can be enclosed with this fencing is then 54.67 feet × 54.67 feet or approximately 2,989.45 square feet.