Competition · AMC preparation · step 4 of 4

AMC 8 · 2006 · #21

Grade 5 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

volume-rectangular-prismarea-rectangleslinear-equations-one-var identify-subproblemsconvert-to-algebra ↑ Prerequisites: volume-rectangular-prismarea-rectangles

📏 Medium solution 💡 3 insights

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Problem

A rectangular aquarium has a base of 100 cm × 40 cm and a height of 50 cm. It is filled with water to a depth of 37 cm. A rock of volume 1000 cm³ is dropped in and fully submerged. By how many centimeters does the water level rise?

Pick an answer.

(A)

0.25

(B)

0.5

(C)

(D)

1.25

(E)

2.5

AMC 8 2006 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Create a Physical Representation

The story is physical: a rock goes in, the water has nowhere to go but up. Tool #10 (Create a Physical Representation) makes the key insight concrete — picture the rock sliding in and shoving an equal volume of water upward. That displaced water spreads across the same rectangular base, forming a thin slab whose volume equals the rock's volume. Tool #7 (Identify Subproblems) splits the work into two clean pieces: (i) compute the base area, (ii) divide the rock's volume by that area to get the slab's thickness, which is the rise. The given numbers 37 cm and 50 cm are deliberately included to confirm the rock fits and the water does not overflow — they do not enter the calculation.

1STEP 1

Compute the base area

Find the base area — the aquarium's 4000 cm² rectangle stays the same at every water level.

Base area = 100 × 40 = 4000 cm²

2STEP 2

Use the displacement idea

The submerged rock shoves up its own volume of water as a thin slab, so the slab's volume is 1000 cm³.

Slab volume = rock volume = 1000 cm³

3STEP 3

Apply the volume formula

Apply the rectangular-prism rule to the slab: volume = base area × thickness, where the thickness is the rise h.

Slab volume = Base area × h → 1000 = 4000 × h

4STEP 4

Solve for the height

Divide both sides by 4000 to get the rise: h = $\frac{1000}{4000}$ = 0.25 cm, choice (A).

h = 1000/4000 = 1/4 = 0.25 cm → (A)

Answer

0.25

Sanity check the size. The base area 4000 cm² is huge compared with the rock's volume 1000 cm³, so spreading 1000 over 4000 should give a thin film — and indeed 0.25 cm (a quarter of a centimeter) is tiny. Also, the new water level would be 37 + 0.25 = 37.25 cm, well below the 50 cm aquarium height, so nothing overflows — the given height and initial depth are red herrings the problem includes to confirm the setup, not to enter the formula. The other answer choices come from likely mistakes: 0.5 from using 50 × 40 = 2000 instead of 100 × 40, 1 and 1.25 from miscomputing the area, 2.5 from

\frac{1000}{400}

💡Key takeaway

A submerged rock pushes up its own volume of water, which spreads across the aquarium's base as a thin slab — divide the rock's volume by the base area and you have the rise.

Compute the base area
Use the displacement idea
Apply the volume formula
Solve for the height

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Compute the base area

Use the displacement idea

Apply the volume formula

Solve for the height

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