A medieval text (Maimonides's commentary to chapter 2 of Eruvin in my retranslation from the Hebrew) discusses a rectangle whose area is $5000$ square cubits. It reads in relevant part:
… that the length should not exceed the width except to such an extent that the diagonal is double the width. Then the length of the region will be $93\frac1{27}$…. All these reckonings are approximate….
"The diagonal is double the width" amounts to $\sqrt{L^2+W^2}=2W$ of course. The length $L$ is then $\sqrt{5000\sqrt3}\approx93.0605$, between $93\frac1{17}$ and $93\frac1{16}$.
Where might he have gotten $1/27$ from?
Note that I'm not asking "How could he be so far off?". I'm asking specifically about the number $93\frac1{27}$ and why he may have that as a solution as opposed to, say, $93\frac1{24}$ or $93\frac1{30}$.