Question 721573
Easy to answer with only a little bit of thinking and without writing anything (except for squaring a number in your head).

Cut the 68 into four equal parts and then square the result.


A more analytical way to solve:


The perimeter is already set at 68 feet.  The question asks, what is the largest area for this rectangular region, based on that given perimeter.  Using x and y for length and width, we have these:


{{{A=xy}}} and {{{2x+2y=68}}}.  A is for AREA.  Using the perimeter equation, 
{{{2y=-2x+68}}}
{{{y=-x+34}}}
Substituting this into A equation,
{{{A=x(-x+34)}}}
{{{highlight(A=-x^2+34x)}}}, which shows that A has a maximum, which would be at the vertex of the graph of A.


Find the vertex!  Put A(x) into standard form, and read the vertex point from standard form equation function A(x).
{{{A(x)=(-1)(x^2-34x+(34/2)^2)-(-1)(34/2)^2}}}
{{{highlight(A(x)=(-1)(x-17)^2+17^2)}}}


Vertex is at (17,289).  This will obviously be a SQUARE SHAPE.  17+17+17+17=68.
Note that a rectangle has its maximum area when it is a square.  This is why one could solve this problem almost entirely in ones head.