Problem 21 · 2006 AMC 8
Easy
Geometry & Measurement
displacement-volume
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. The aquarium is filled with water to a depth of 37 cm. A rock with volume 1000 cm3 is then placed in the aquarium and completely submerged. By how many centimeters does the water level rise?
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Answer: A — 0.25 cm.
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Hint 1 of 2
The submerged rock pushes water up. That extra water forms a thin flat slab sitting on top, with the rock's volume but the tank's base. Picture that slab.
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Hint 2 of 2
A slab's volume = base area × height. You know the slab's volume (= rock volume) and the base, so the rise is volume ÷ base area. The starting depth and tank height are decoys.
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Approach: the rise is a thin slab with the rock's volume
- Submerging the rock displaces exactly 1000 cm³ of water, which spreads across the full base as a thin layer of rise.
- That layer is a box: base 100 × 40 = 4000 cm², volume 1000 cm³. Its height (the rise) = 1000 ÷ 4000 = 0.25 cm.
- Why the 37 cm and 50 cm don't matter: the rise depends only on how much volume you add and how wide the tank is — not on the current depth (as long as the rock stays submerged and the water doesn't overflow). Sanity check: 0.25 cm of rise × 4000 cm² = 1000 cm³, exactly the rock. Spotting the decoy numbers is half the battle.
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