It's easy. Believe me

We have to use only 200 meters of fencing to make a rectangular enclosure with maximum area. This means that the perimeter of this enclosure is equal to 200 meters, or 2 (height+width)=200. We also know that the area of a rectangle is equal to height width. We can then mount a system with two linear equations:

2 (h+w)=200 (1) Area=h w (2)

Dividing both members of the first equation by 2, we have that:

h+w=100 (3)

Isolating h on the first equation by subtracting both members of the equation by w, we have:

h=100-w (4)

Substituting h by (100-w) in the secundário equation, we can say that:

Area=(100-w)*100w -w^2 (5)

This is a decreasing quadratic function, and it has a maximum value, which is the vertex of its parabola, that we want to find. The formula to find it is:

x=-b/(2*a) (6)

Were b and a are values of the quadratic function model:

f(x)=ax^2+bx+c (7)

Applying (6) on equation 5, we get:

w=-(100/(2*(-1)))=50 (8)

Which is the maximum value of w. As we know by (4) that h=100-w, we can substitute w for 50, which means that h=100-50=50. Now we can see that our rectangle is actually a square, because its height is equal to its width, and its perimeter is 50*4=200. Now, as we know the size of the sides that make our enclosure have the maximum area possible, the area of the rectangle is:

A=50^2= $\boxed{2500}$ (9) $_\square$

P.S.: This solution does not use calculus, although you can use it to solve this problem. If you have and question, correction, or just have something to say, i would like to hear you.