Your sentence is dramatically better than theirs because you're referring to the length dimension of the rope
and the track. "What length of rope does it take to go around a quarter-mile wide track?" <- using the author's mistake in your sentence. There's not enough information in the sentence to answer correctly.
The correct word for a measurement of wideness is width.
Length [NOUN]:
1.
the longest extent of anything as measured from end to end:
the length of a river.
2.
the measure of the greatest dimension of a plane or solid figure.
It's pretty unambiguously wrong. You can back into what the author meant without ambiguity, but that doesn't mean the author used the correct word.
