The area is based on the length and width of the fenced area, which depends on the fencing material he has.
Since he only has to fence 3 sides of the field, the perimeter of the field is width + length + width, or P = 2W + L
We know that the perimeter will equal 360 meters, so:
360 = 2W + L
360 - 2W = L
The area of a rectangle is width * length, or A = W*L
Since L = 360 - 2W we can rewrite our area equation as
A = W (360-2W)
If we multiply this out we get
A = 360W - 2W2
If we rearrange this equation we get
2W2 -360W + A = 0
Now we have a quadratic equation. The max/min of a quadratic is at the vertex. To find the x axis point of the vertex we use the equation
X = -b/2a
Which in this case will be the width of the field.
Width = -(-360)/2(2) = 360/4 = 90
So our width at the maximum area will be 90 meters. You can use the other equations to find the length and area of the field from there.