A glass box cm, closed on all six sides, is partially filled with colored water. When the box is placed on one of its sides, the water level is cm above the table. If the box is placed on one of its 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 sides, the volume of water = . If represents the level of water above the table when the box is placed on one of its sides, the same volume . Hence the water level cm.

**Answer**: 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 , or 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 , or depending on which side it is placed on. Since the water level equals when the box height equals , the ratio of water level to box height is . So when the box is placed on one of its sides, the water level equals cm.

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Retired high school math teacher.