Answer
250 yards by 500 yards; maximum area $125,000\ yard^2$
Work Step by Step
Step 1. Draw a diagram as shown in the figure. Assume one side normal to the stream is $x$. As the total length of the fence is 1000 yards, then the unknown side will be $1000-2x$
Step 2. The area is given as $A=x(1000-2x)=-2x^2+1000x=-2(x^2-500x+250^2)+2\times250^2=-2(x-250)^2+125,000$
Step 3. We can find the maximum of the area at $x=250\ yards$ and $A=125,000\ yard^2$. The other side can be found as $1000-2(250)=500\ yards$
Step 4. Thus, the dimensions of the rectangle are 250 yards by 500 yards with a maximum area $125,000\ yard^2$
