Question 697679
{{{ x }}} is the length of the side parallel to the river.
Each of the sides that are perpendicular to the river 
are {{{ ( 526 - x ) / 2 }}}
Let {{{ A }}} = area of enclosed portion
{{{ A = x*( 526 - x ) / 2 }}}
{{{ A = ( -x^2 + 526x ) / 2 }}}
{{{ A = (-1/2)*x^2 + 263x }}}
This is a parabola which has a maximum because the
coefficient of the {{{ x^2 }}} term is negative and the {{{ x }}}
term is positive. The maximum is at {{{x[max] =  -b/(2a) }}}
when the equation has the form {{{ A = ax^2 + bx + c }}}
{{{ a = -1/2 }}}
{{{ b = 263 }}}
{{{ -b/(2a) = -263 / ( 2*(-1/2) ) }}}
{{{ -b/(2a) = 263 }}}
263 meters answer
This is the x-co-ordinate of the maximum area, which is the
length of fencing parallel to the river.
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If you want the maximum Area, just plug this back into the equation
{{{ A[max] = (-1/2)*263^2 + 263*263 }}}
{{{ A[max] = -69169/2 + 69169 }}}
{{{ A[max] = 34584.5 }}} m2
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It's easy to test this result. if {{{ x }}} is slightly more than or less
than {{{ 263 }}}, the area should go down slightly also
If I say {{{ x = 262.8 }}}, then
{{{ A = (-1/2)*262.8^2 + 262.8*263 }}}
{{{ A = (-1/2)*69063.84 + 69116.4 }}}
{{{ A = -34531.92 + 69116.4 }}}
{{{ A = 34584.48 }}} It went down
You can check for {{{ x = 263.2 }}}
The area should also go down by roughly
the same amount.