To find the greatest possible area that can be enclosed by the rectangular enclosure, we need to maximize the area while using all 800m of fencing. Let's denote the length of the rectangle as L and the width as W.
To start, let's consider the perimeter of the rectangle. The perimeter is the sum of all four sides, which in this case is 800m:
Perimeter = 2L + 2W = 800m
Now we can solve this equation for either L or W in terms of the other variable. Let's solve it for L:
2L = 800m - 2W
L = (800m - 2W) / 2
Next, we need an equation to represent the area of the rectangle. The area of a rectangle is given by length multiplied by width:
Area = L * W
Now substitute the expression for L into the area equation:
Area = [(800m - 2W) / 2] * W
Simplify the equation:
Area = (800mW - 2W^2) / 2
Now we have the area equation in terms of a single variable, W. To find the maximum area, we need to determine the value of W that maximizes this equation. We can achieve this by finding the vertex of the parabola.
The maximum value of the area occurs at the vertex, which can be found using the formula:
W_vertex = -b / (2a)
In this case, a = -2 and b = 800m. Plug these values into the formula:
W_vertex = -800m / (2 * -2)
W_vertex = -800m / -4
W_vertex = 200m
So the width of the rectangle that maximizes the area is 200m.
To find the length, substitute this value of W into the equation for L:
L = (800m - 2(200m)) / 2
L = (800m - 400m) / 2
L = 400m / 2
L = 200m
Therefore, the dimensions of the rectangle that encloses the greatest possible area using 800m of fencing are 200m x 200m, and the corresponding maximum area is 40,000 square meters.