solution: Let the width of the rectangle be $w,$ and let the radius of each semicircle be $r.$
  
  [asy]
  unitsize(1 cm);
  
  filldraw((0,0)--(3,0)--(3,2)--(0,2)--cycle,lightgreen);
  draw((0,0)--(3,0),linewidth(2*bp));
  draw((0,2)--(3,2),linewidth(2*bp));
  draw(arc((3,1),1,-90,90),linewidth(2*bp));
  draw(arc((0,1),1,90,270),linewidth(2*bp));
  
  label("$w$", (1.5,0), S);
  label("$r$", (3,1/2), E);
  dot((3,1));
  [/asy]
  
  Then the length of the track is $2w + 2 \pi r = 400,$ so $w + \pi r = 200.$  By AM-GM,
  \[200 = w + \pi r \ge 2 \sqrt{w \pi r},\]so $\sqrt{w \pi r} \le 100.$  Then $w \pi r \le 10000,$ so
  \[wr \le \frac{10000}{\pi}.\]Then the area of the field, $2wr,$ must satisfy
  \[2wr \le \frac{20000}{\pi}.\]Equality occurs when $w = 100$ and $r = \frac{100}{\pi},$ so the largest possible area is $\boxed{\frac{20000}{\pi}}.$
prompt.content: <|im_start|>system
  You are a helpful assistant.<|im_end|>
  <|im_start|>user
  A rectangular field is enclosed in an athletic track, as shown below.  The track consists of two edges of the field, and two semicircles.  The length of the track is 400 meters.  What is the largest possible area of the field, in square meters?
  
  [asy]
  unitsize(1 cm);
  
  filldraw((0,0)--(3,0)--(3,2)--(0,2)--cycle,lightgreen);
  draw((0,0)--(3,0),linewidth(2*bp));
  draw((0,2)--(3,2),linewidth(2*bp));
  draw(arc((3,1),1,-90,90),linewidth(2*bp));
  draw(arc((0,1),1,90,270),linewidth(2*bp));
  [/asy]
  Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
  <|im_start|>assistant
  
prompt.role: user
extra_info.answer: \frac{20000}{\pi}
extra_info.question: A rectangular field is enclosed in an athletic track, as shown below.  The track consists of two edges of the field, and two semicircles.  The length of the track is 400 meters.  What is the largest possible area of the field, in square meters?
  
  [asy]
  unitsize(1 cm);
  
  filldraw((0,0)--(3,0)--(3,2)--(0,2)--cycle,lightgreen);
  draw((0,0)--(3,0),linewidth(2*bp));
  draw((0,2)--(3,2),linewidth(2*bp));
  draw(arc((3,1),1,-90,90),linewidth(2*bp));
  draw(arc((0,1),1,90,270),linewidth(2*bp));
  [/asy]
extra_info.split: test