Propositiones ad Acuendos Juvenes/Problems/27 - De Civitate Quadrangula

Propositiones ad Acuendos Juvenes by Alcuin of York: Problem $27$

A four-sided town measures $1100$ feet on one side

and $1000$ feet on the other side;

on one edge $600$

and on the other edge $600$.

I want to cover it with roofs of houses,

each of which should be $40$ feet long

and $30$ feet wide.

How many dwellings can I make there?

The two long sides add up to $2100$ feet.

Similarly the two short sides add up to $1200$ feet.

Then the half of $1200$ is $600$, and the half of $2100$ is $1050$.

Because each house is $40$ feet long and $30$ feet wide:

take the $40$th part of $1050$ which is $26$

and the $30$th part of $600$ which is $20$.

$20$ times $26$ is $520$.

That is how many houses can be built.

The Egyptian Formula for Area of Quadrilateral is being used to estimate the total area of the town.

Alcuin makes no indication of actually trying to fit the houses into the town.

Because the shape of the town is not actually rectangular, the exercise is not trivial.

David Singmaster reports that he has tried, but cannot actually get any more than $519$ in, and that is by turning the houses in either direction.