## Water in Glass Box

A glass box $7\times 12\times 18$ cm, closed on all six sides, is partially filled with colored water. When the box is placed on one of its $7\times 12$ sides, the water level is $15$ cm above the table. If the box is placed on one of its $7\times 18$ sides, what will be the water level above the table, in centimeters?
Source: NCTM Mathematics Teacher, August 2006

Solution
When the box is placed on one of on its $7\times 12$ sides, the volume of water = $7\times 12\times 15$. If $x$ represents the level of water above the table when the box is placed on one of its $7\times 18$ sides, the same volume $7\times 12\times 15=x\times 7\times 18$. Hence the water level $x=10$ cm.

Answer: $10$ cm

Alternative solution
If we filled the box completely full with water, the box will look full no matter how we set it on the table. The height of the water level equals the height of the box $7,12$, or $18$ cm depending on which side it is placed on. Similarly, if we filled the box half full, the box will look half full no matter how we set it on the table. The height of the water level equals half the height of the box $7/2,12/2$, or $18/2$ depending on which side it is placed on. Since the water level equals $15$ when the box height equals $18$, the ratio of water level to box height is $15/18=5/6$. So when the box is placed on one of its $7\times 18$ sides, the water level equals $(5/6)12=10$ cm.