The maximum area of a rectangle given a constant perimeter is achieved when the rectangle is a square.
We can approach this problem by first considering the nature of a square and a rectangle. A rectangle's perimeter is the sum of its sides or `2l + 2w` where `l` is the length and `w` is the width. Now in a square, the length equals the width (since all sides are of equal measure) so you can substitute `w` with `l`.
This effectively results in the equation `2l + 2l = 4l = Perimeter` where `Perimeter` is the given perimeter, which in this case is 150 feet.
Isolating the length `l`, we divide the perimeter by 4: `l = Perimeter/4`,
Substituting the perimeter value, we get `l = 150/4 = 37.5` feet. Since the rectangle is a square, the width is also 37.5 feet.
The maximum area can be found by multiplying the length and the width of the square. That is, `Area = length * width`. We then substitute the values we found: `Area = 37.5 * 37.5 = 1406.25` square feet.
So, with a perimeter of 150 feet, the dimensions of this region that will enclose the maximum area, a square, will be 37.5 feet by 37.5 feet, and the maximum area will be 1406.25 square feet.