Question 914250
Does more information come with this question?  Six separate corrals, not connected to each other?  


Maybe x and y, dimensions of each corral, all corrals congruent.  ASSUMING rectangle shape.


The fencing is the sum of the perimeters.  {{{6(2x+2y)=840}}}
{{{12(x+y)=840}}}
{{{x+y=70}}}


Total area, {{{A=6xy}}}


Substitute for either variable, choosing y, in the area equation:
{{{A=6x(70-x)}}}
Do some steps,
{{{A=-6x^2+420}}}, but maybe not necessary in that form.


A is a parabola opening downward having a vertex as a maximum point, which is what you want to solve for.  Find the roots!
{{{A=0=6x(70-x)}}}
{{{-6x(x-70)=0}}}
Roots are 0 and 70.
The vertex will be for the exact middle of these roots, which will be {{{highlight(x=35)}}}.
Notice that according to the earlier found equation from perimeter, x+y=70, which means that also {{{highlight(y=35)}}}.
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Each corral will be a square with side 35 feet.


Based on the questions part of what you posted, some of the description is missing and so maybe the answer I found does not fit what you really were given.  I could only assume that your six corrals would be rectangles and not connected to each other.  Otherwise, give the complete problem description.