Final answer:
Serenity can calculate the possible lengths and widths for her field by using the area and perimeter of the rectangle, leading to a quadratic equation solving for the width, and then finding the corresponding length.
Step-by-step explanation:
To find the possible dimensions of length and width for Serenity's field, which is a rectangular plot of land with three sides fenced and one side along the riverbank, we can use the perimeter and area formulas for a rectangle. Serenity has 55 meters of fencing available for the three sides. Let's denote the length of the rectangle (parallel to the river) as L meters and the width (the two sides perpendicular to the river that will be fenced) as W meters.
Since the total length of the fence available is 55 meters, we know that the sum of the width sides and one of the length sides must be equal to this, as one length side does not need fencing. So:
L + 2W = 55
The area A of the rectangle is given as 342 m², so:
A = L × W = 342
To find the possible dimensions, we can isolate one of the variables from the first equation and substitute it into the area equation. Isolating L, we get:
L = 55 - 2W
Substituting into the area equation:
342 = (55 - 2W) × W
Rearrange the equation:
0 = 2W² - 55W + 342
Solving this quadratic equation will give us the possible values for W, and consequently, we can use these values to find the corresponding values for L. These pairs will be the possible dimensions for the length and width of the land.