AMC 10 · 2021 · #10

Grade 8 geometry-3d

Archetype Compound Area by Decomposition / Difference └ Sub-archetype Polygon Triangulation

similar-trianglesratio-proportionvolume-cylinder identify-subproblemseasier-related-problem ↑ Prerequisites: similar-triangles

📏 Medium solution 💡 3 insights 📊 Diagram

Problem

Two apex-down cones hold the same volume of liquid. The liquid surface in the narrow cone is a circle of radius 3 centimetres; in the wide cone it is a circle of radius 6 centimetres. One identical marble of radius 1 centimetre is dropped into each cone and both fully sink. Find the ratio of the rise in the narrow cone to the rise in the wide cone.

Pick an answer.

(A)

1:1

(B)

47:43

(C)

2:1

(D)

40:13

(E)

4:1

How to solve

Strategy Easier Related Problem

Tool #9 (Easier Problem) does the heavy lifting: imagine the cones are SO TALL that near the liquid surface they look like cylinders of radius 3 and 6. Then equal added volume V_m raises the narrow cylinder by V_m / (π · 3²) and the wide cylinder by V_m / (π · 6²) — ratio 36/9 = 4. Tool #1 (Draw) makes the cone-vs-cylinder picture explicit. Tool #7 (Subproblems) cleans up the rigorous version: same initial volume implies the narrow cone's height is 4 × the wide cone's height, and that same factor of 4 then reappears in the rise ratio. Tool #3 (Eliminate) confirms (E) against the five choices.

1STEP 1

Describe each cone

In each cone the radius is proportional to the height.

Narrow: r = 3/h_n · h. Wide: r = 6/h_w · h.

2STEP 2

Relate the two heights

Equal volumes fix the height ratio.

1/3π (3)² h_n = 1/3π (6)² h_w → h_n = 4 h_w

3STEP 3

Why the surface alone misleads

Using only the surface gives a different answer.

Δ_n^cyl/Δ_w^cyl = (V_m / (π · 9))/(V_m / (π · 36)) = 36/9 = 4

4STEP 4

Write the volume increase

Write each cone's volume increase.

H_n³ - h_n³ = (V_m h_n²)/3π, H_w³ - h_w³ = (V_m h_w²)/12π

5STEP 5

Cancel the marble

The marble's volume cancels.

Δ_n/Δ_w = (H_n - h_n)/(H_w - h_w) = h_n/h_w = 4

6STEP 6

Read the ratio

The ratio is 4 to 1.

Δ_n : Δ_w = 4 : 1 → (E)

Answer

4:1

Two ways to feel that 4:1 is right. (a) The cylinder approximation (Tool #9) already gave exactly 4:1 — and the marble adds the same volume to each cone, so wider rim = smaller rise. The wide rim is twice the narrow rim, and area scales as radius squared (6² / 3² = 4), so the wide cone needs 4 × less rise. (b) The narrow cone is four times TALLER for the same volume, and the rise scales proportionally with the original height — so 4:1 again. Both arguments converge, and the answer does NOT depend on the marble's actual volume — it could be any small object and the ratio stays 4:1.

💡Key takeaway

This AMC 12 problem only needs Grade 8 volume formulas and proportional reasoning you already know! Same liquid volume + a wider top means the narrow cone is 4 × taller. Drop the SAME marble into both — the wider rim spreads the rise out over 4 × the cross-section area, so the narrow cone's level jumps up 4 × as much. Ratio 4:1, answer (E).