A farmer is building a fence to enclose a rectangular area against an existing wall. Three of the sides will require fencing and the fourth wall already exists. If the farmer has 252 feet of fencing, what is the largest area the farmer can enclose?

To maximize the area using 252 feet of fencing against an existing wall, we can utilize the formula for the area of a rectangle, which is length times width (A = L × W). Given that the wall serves as one of the long sides, the farmer will only need to fence the other three sides, resulting in the equation: L + 2W = 252. By rearranging the formula to express L in terms of W (L = 252 - 2W) and substituting into the area equation, we find that maximizing the area leads us to set W to 63 feet and L to 126 feet. This provides an enclosed area of 7,938 square feet! Finding the optimal dimensions can sometimes be tricky! One common mistake is miscalculating the perimeter constraint. Remember, since one side is against the wall, only three sides need fencing. Always double-check that you’re allocating the fencing correctly to avoid running short or overestimating your area. It’s a great practice to lay out what you have before diving into complex calculations!