Let
x--------> the length side of the rectangular plot (assume side along the river)
y------> the width side of the rectangular plot
we know that
the perimeter of the rectangular plot is equal to
\(P=x+2y\)
\(P=7,000\ m\)
so
\(7,000=x+2y\)
Clear variable y
\(y=(7,000-x)/2\) -------> equation \(1\)
The area of the rectangular plot is
\(A=x*y\) ------> equation \(2\)
substitute equation \(1\) in equation \(2\)
\(A=x*[(7,000-x)/2]\)
\(A=x*[(7,000-x)/2]\\\\ A=3,500x-0.5x^{2}\)
we know that
To find the larger area that can be enclosed------> Find the vertex of the quadratic equation
The quadratic equation is a vertical parabola open down
so
the vertex is a maximum
using a graphing tool
see the attached figure
the vertex is the point \((3,500,6,125,000)\)
that means
For \(x=3,500\ m\)
the largest area is \(6,125,000\ m^{2}\)
Find the value of y
\(y=(7,000-3,500)/2=1,750\ m\)
the dimensions of the rectangular plot are
\(Length=3,500\ m\\Width=1,750\ m\)
therefore
the answer is
The largest area that can be enclosed is \(6,125,000\ m^{2}\)
