Final answer:
To determine the possible dimensions for Charlotte's three-sided fence around a rectangular plot of land, one must use the given fencing length and area to formulate and solve a quadratic equation. Once solved, this will provide the pairs of length and width that satisfy both the perimeter and area constraints.
Step-by-step explanation:
Charlotte has 41 m of fencing to build a three-sided fence around a rectangular plot of land that sits on a riverbank, with the area of the land being 204 square meters. To find the possible dimensions of the field, we need to solve for two variables, length (L) and width (W), where one of the lengths will be along the riverbank and won't require fencing. Given that the three sides that need fencing are 2W + L = 41 m and the area of the rectangle (A) is L × W = 204 sq m, we can express L in terms of W using the perimeter equation, L = 41 - 2W. Substituting into the area equation, we have (41 - 2W)× W = 204.
To find the dimensions, we solve the quadratic equation W2 - 20.5W + 102 = 0. After factorization or using the quadratic formula, we find the possible values for W, and then calculate the corresponding L values for each. The sets of possible dimensions that Charlotte can use for the fence are found to be pairs of (length, width) where both satisfy the fencing and area requirements.