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Mathematics-Online problems:

Problem of the week

Having become insolvent, a company is liquidated and its property in need of rehabilitation is transferred to the four municipalities $ A$, $ B$, $ C$, and $ D$ neighboring these premises. The municipalities decide to divide up and restructure a quadratic area of 1km$ ^2$ altogether (left drawing), which encloses the premises in need of rehabilitation (shaded square). After restructuring the area, the borders should be straight lines (right drawing).

The clean-up costs totalling 9 million Euro are to be divided among the municipalities according to the parts of the premises they obtain. Since the municipalities $ A$ and $ B$ are not able to bear any clean-up costs due to their deplorable financial situation, the size of their part of the whole area should stay unchanged.


Determine the areas of all estates before and after restructuring, the lengths of the new borderlines, and the shares of the clean-up costs the two municipalities incurred.


$ A:$          $ B:$
$ C:$          $ C':$
$ D:$          $ D':$
(Areas in square meters)

Lengths of borderlines:
$ \,\,x:$          $ \,\,\,y:$

Clean-up costs:
$ \,$Share of $ C:$      $ \,$Share of $ D:$
(Amounts in euros)


[solution to the problem of the previous week]