Question 1001420
A square shape will give maximum area.  I do not give that analysis here.  The perimeter of this would be  {{{120+10}}}, to include both the fence and the gate(unfenced).  Cut the 130 feet into four equal parts, and this is the side of the square garden.


{{{32&1/2}}} feet square.



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MAXIMIZE AREA


Dimensions are x and y.
The entire perimeter of the garden is length_of_fencing PLUS 10 feet for gate; so this means the perimeter of the garden is  {{{120+10=130}}} feet.


Two basic equations are needed.
{{{system(2x+2y=130,A=xy,A=AreaOfGarden)}}}.


Use these to make a function A dependent on just ONE of the variables, either x or y.


{{{2x+2y=130}}}
{{{x+y=65}}}
{{{y=65-x}}}
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{{{A=xy}}}
{{{highlight(A=x(65-x))}}}------Area A is a function of the dimension, x, a quadratic function, and you can look for the MAXIMUM.  You know there is a maximum because the coefficient on {{{x^2}}} will be negative...
The max area will occur in the exact middle of the roots or zeros or x-intercepts of {{{A(x)}}}.
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Solve for the x-intercepts.
{{{A(x)=0=x(65-x)}}}


The roots are 0 and 65.
The value for x exactly in the middle is  {{{(0+65)/2=65/2=32&1/2}}}.


WHAT IS THE VALUE FOR y FOR THIS VALUE OF x ?
WHAT IS THE AREA AT THIS VALUE OF x ?