Question 551427
The fence will go a distance x perpendicularly away from the banks of the river at two places, with a stretch of fence in between, parallel to the river, and measuring 1000-2x meters.
The area fenced in is
{{{A=x(1000-2x)=-2x^2+1000x}}}
The quadratic function
{{{A(x)=-2x^2+1000x}}} , like all quadratic functions {{{f(x)=ax^2+bx+c}}}
with {{{a<0}}}, has a maximum (and line of symmetry) at {{{x=-b/2a}}}.
In this case, the maximum and line of symmetry are at {{{x=-1000/2(-2)=-1000/-4=250}}},
So, 250 meters will be the width of the rectangular plot.
The length of the rectangular plot (along the river) will be (in meters)
{{{1000-2x=100-2*250=1000-500=500}}}
and the area (in square meters) will be
{{{A(250)=125000}}}