Question

Peter's farm has 160 meters of fencing, and he wants to fence a rectangular field thatborders a straight river. He needs no fence along the river side. Find the largest areaof Peter's farm that can be fenced.

Answer 1

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,

$P=W+W+L=2W+L$

And this is equal to 160 m. Solving for L,

$160=2W+L\text{ }\Rightarrow\text{ }L=160-2W$

The area of this field is,

$A=WL$

Replace L with the expression we found from the perimeter,

$A=W(160-2W)=160W-2W^2$

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,

$f(x)=ax^2+bx+c\text{ }\Rightarrow\text{ }x_{vertex}=\frac{-b}{2a}$

In this case, x is W, a = -2, and b = 160, so the width of the field for the maximum area is,

$W_{vertex}=\frac{-160}{2(-2)}=\frac{160}{4}=40m$

And the length when W = 40 is,

$L=160-2W=160-2\cdot40=160-80=80m$

And the area is,

$A=WL=40m\cdot80m=3200m^2$

Hence, the largest area of Peter's farm that can be fenced is 3200 square meters.