AMC 10 · 2007 · #3

Grade 5 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Sum of Primitive Solids
volume-rectangular-prismarea-rectanglesidentify-subproblems identify-subproblems ↑ Prerequisites: volume-rectangular-prism
📏 Short solution 💡 2 insights
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Problem
An aquarium with a rectangular base 100 cm by 40 cm and height 50 cm holds water 40 cm deep. A solid brick measuring 40 cm by 20 cm by 10 cm is set on the bottom. Find how many centimeters the water surface rises.

Pick an answer.

(A)
0.5
(B)
1
(C)
1.5
(D)
2
(E)
2.5

AMC 10 2007 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Draw a Diagram

A quick cross-section drawing (Tool #1, Draw a Diagram) shows the key fact: the brick sits fully under the water and shoves aside a chunk of water equal to its own volume, and that water can only go up, forming a thin slab across the entire base. Once that picture is clear, the number splits into two clean subproblems (Tool #7): first the brick's volume, then the height of a slab with that volume spread over the tank's base. Tool #8 (Analyze the Units) confirms the last step, since a volume in cm³ divided by a base area in cm² must give a height in cm.

1STEP 1

See what the brick does to the water

Side view: the brick is only 10 cm tall but the water is 40 cm deep, so it is fully submerged and the water it shoves aside can only go up.

2STEP 2

Find the brick's volume

The brick is a box, so its volume is 40×20×10=8000 cm³ — exactly the amount of water shoved upward.

40×20×10=8000 cm³
3STEP 3

Spread that volume across the base

That slab covers the whole base, 100×40=4000 cm², so 4000h=8000 and the water rises h=2 cm — choice (D).

4000 × h = 8000 → h = 8000/4000=2 cm → (D)
Answer
2
The answer is small and positive, which fits — a modest brick should nudge a wide tank's water up only a little. Check the size: the brick's footprint is 40×20=800 cm², one fifth of the tank's 4000 cm² base, and the brick is 10 cm tall, so spreading that 10 cm of solid over a base five times wider gives about 10×800/4000=2 cm — matching. Also the tank does not overflow: the level goes from 40 cm to 42 cm, still under the 50 cm rim, so the whole brick really does stay submerged and the reasoning holds. Answer 2 cm is (D).
💡Key takeaway

A sunk object pushes up its own volume of water, so spread that volume over the tank's base to find how high the water climbs.

  • See what the brick does to the water
  • Find the brick's volume
  • Spread that volume across the base

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