Question 36209
Let the length of the rectangular patio be x ft
Perimeter of the rectangular patio = 400 ft 
The width of the rectangular patio would be 1/2(400-2x) = 200-x
Area = Length * Width = x(200-x)
We need to maximize f(x) = x(200-x) to achieve the goal of the problem.
f(x) is quadratic function for Area, which represents a parabola opening downward (because the co-efficient of x^2 is negative).
Maximum of f(x) is reached at the vertex (because it opens downward)
x-Coordinate of the vertex = -b/2a (for ax^2+bx+c);
f(x) = -x^2+200x; So, a = -1, b=200, c=0
(-b/2a) = 100;
So, maximum area can be obtained if the length is 100 feet;
The width will be 1/2(400-200) = 100 feet;
Answer - Dimension of the patio for maximum area will be 100 feet X 100 feet