First, let's find a function that will give us the area with for the dimensions.
Since the area is rectangular, let two sides be x and the other be y:
This means that the sum of these three sides have to be equal to 1000 meters:
[tex]\begin{gathered} x+y+x=1000 \\ y+2x=1000 \\ y=1000-2x \end{gathered}[/tex]Also, the area of the rectangle is:
[tex]\begin{gathered} A(x,y)=xy \\ A(x)=x(1000-2x) \\ A(x)=1000x-2x^2 \\ A(x)=-2x^2+1000x \end{gathered}[/tex]This is a quadratic function and, since its leading coefficient is negative, it has a maximum value at its vertex. The x value of the vertex corresponds to the x variable of the function, and the y value corresponds to the maximum value of the function A(x).
The x coordinate of the vertex of a quadratic function is:
[tex]\begin{gathered} x_V=-\frac{b}{2a} \\ x_V=-\frac{1000}{2(-2)} \\ x_V=-\frac{1000}{-4} \\ x_V=250 \end{gathered}[/tex]So, the value of x for which the field has maximum area is 250, which means that the other side of the field is:
[tex]\begin{gathered} y=1000-2x \\ y=1000-2\cdot250 \\ y=1000-500 \\ y=500 \end{gathered}[/tex]So, the field will have maximum area when its dimensions are 250 meter x 500 meters.
The y value of the vertex can be calculated inputting the x value into the funciton, so:
[tex]\begin{gathered} A_{\max }=A(x_V)=-2(250)^2+1000\cdot250 \\ A_{\max }=-2\cdot62500+250000 \\ A_{\max }=-125000+250000 \\ A_{\max }=125000 \end{gathered}[/tex]So, the maximum are of the field will be 125000 squared meters.
