ANSWER
3200 m²
EXPLANATION
Peter wants to build a fence around a rectangular field, except for one side because the river is there,
So, the total perimeter of the fence is,
[tex]P=W+W+L=2W+L[/tex]And this is equal to 160 m. Solving for L,
[tex]160=2W+L\text{ }\Rightarrow\text{ }L=160-2W[/tex]The area of this field is,
[tex]A=WL[/tex]Replace L with the expression we found from the perimeter,
[tex]A=W(160-2W)=160W-2W^2[/tex]The area is given by a quadratic function whose leading coefficient is negative, which means that the graph is a downward parabola and, therefore, the vertex is the maximum value of the area.
The x-coordinate of the vertex is given by,
[tex]f(x)=ax^2+bx+c\text{ }\Rightarrow\text{ }x_{vertex}=\frac{-b}{2a}[/tex]In this case, x is W, a = -2, and b = 160, so the width of the field for the maximum area is,
[tex]W_{vertex}=\frac{-160}{2(-2)}=\frac{160}{4}=40m[/tex]And the length when W = 40 is,
[tex]L=160-2W=160-2\cdot40=160-80=80m[/tex]And the area is,
[tex]A=WL=40m\cdot80m=3200m^2[/tex]Hence, the largest area of Peter's farm that can be fenced is 3200 square meters.
