Answer
Dimensions: ($250$ yd)$\times$ ($500$ yd)
Area: $125,000$ square yards.
Work Step by Step
The area A = (width)$\times$(length),
$A=x(1000-2x)=1000x-2x^{2}$
As a function of x, $A(x)=-2x^{2}+1000x$
is a quadratic function, $A(x)=ax^{2}+bx+c$.
whose graph is a parabola that opens down, since $a=-2$, negative.
Its vertex is a maximum point, which occurs at
$x=-\displaystyle \frac{b}{2a}=-\frac{1000}{2(-2)}=250$ yd.
So, the dimensions of the field of greatest area are
($250$ yd)$\times$ ($1000-2\cdot 250$ yd)
$= $($250$ yd)$\times$ ($500$ yd)
The area of that rectangle is
$A(250)=125,000$ square yards.
