Final answer:
The dimensions of the region that encloses the maximal area are 40.5 feet by 40.5 feet, and the maximum area is 1640.25 square feet. The minimum area is 0 square feet.
Step-by-step explanation:
To find the dimensions of the region that encloses the maximal area, we can use the formula for the area of a rectangle, which is A = length x width. In this case, we know that the farmer has 162 feet of fencing, so the total perimeter of the rectangle is 162 feet.
Let's assume the length of the rectangle is x and the width is y. From the given information, we can set up the equation 2x + 2y = 162 (equation 1), since the perimeter is the sum of all four sides of the rectangle.
We can solve equation 1 for y in terms of x by subtracting 2x from both sides and dividing by 2: y = (162 - 2x) / 2 = 81 - x.
To find the maximum area, we need to find the maximum value of the function A = x * y. Substituting the expression for y from equation 1, we have A = x * (81 - x).
To find the maximum value of A, we can find the vertex of the quadratic function A = x * (81 - x). The x-coordinate of the vertex is given by x = -b/2a, where a = -1 (coefficient of x^2 term) and b = 81 (coefficient of x term). Plugging in these values, we get x = -81/(2*(-1)) = 40.5.
Substituting this value for x back into equation 1, we can find the corresponding value for y: y = 81 - 40.5 = 40.5.
Therefore, the dimensions of the region that encloses the maximal area are 40.5 feet by 40.5 feet, and the maximum area is (40.5)(40.5) = 1640.25 square feet.
The minimum area occurs when one of the dimensions is 0. In this case, the region is just a line segment and does not enclose an area. Therefore, the minimum area is 0 square feet.