Answer
Dimensions: $250\text{ yards}\times 500\text{ yards}$
Area: $125,000\text{ square yards}$
Work Step by Step
Since the total length of the amount of material is $1000$ and two sides of the rectangle have the length $x$, then the third, which will be enclosed, will have the length $1000-2x$.
Now let's consider the area function:
$$A(x)=x(1000-2x)=-2x^2+1000x.$$
We need to find $x$ so that $A$ reaches its maximum.
Bring the function to the vertex form $A(x)=a(x-h)^2+k$:
$$\begin{align*}
A(x)&=-2(x^2-500x)\\
&=-2(x^2-500x+250^2)+2(250^2)\\
&=-2(x-250)^2+125,000.
\end{align*}$$
Identify the constants $a$, $h$, $k$:
$$\begin{align*}
a&=-2\\
h&=250\\
k&=125,000.
\end{align*}$$
Determine the vertex of the function:
$$(h,k)=(250,125,000).$$
The function $A$ is a quadratic function with negative leading coefficient. This means that its graph is a parabola opening downward, which has its maximum in the vertex. Since the vertex is $(250,125,000)$, it means the value of $x$ for which the maximum is reached is $x=250$. The rectangle's sides are:
$$\begin{align*}
x&=250\\
1000-2x&=1000-2(250)=500.
\end{align*}$$
The maximum area is $125,000$ square yards.
