AMC 10 · 2015 · #9

Grade 8 geometry-3d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series
volume-cylinderpercentageratio-proportion convert-to-algebra ↑ Prerequisites: volume-cylinderpercentage
📏 Medium solution 💡 2 insights
Problem
Two right circular cylinders hold the same volume. The second cylinder's radius is 10% larger than the first cylinder's radius. Describe how the two heights compare.

Pick an answer.

(A)
The second height is } 10\% \text{ less than the first.
(B)
The first height is } 10\% \text{ more than the second.
(C)
The second height is } 21\% \text{ less than the first.
(D)
The first height is } 21\% \text{ more than the second.
(E)
The second height is } 80\% \text{ of the first.

AMC 10 2015 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Introduce a Variable

The words give a relationship, not numbers, so Tool #4 (Introduce a Variable) names the two radii and two heights and writes each volume with the same formula. Equal volume then becomes one equation. Tool #13 (Convert to Algebra) does the work inside that equation: the shared π r₁² cancels and 1.1² collapses to a single factor, leaving the two heights compared directly. Finally Tool #3 (Eliminate Possibilities) reads that factor as a percent and matches it to exactly one of the five choices while exposing the 10% trap.

1STEP 1

Name the parts and set volumes equal

Let both cylinders have volume π r² h; equal volumes give π r₁² h₁=π r₂² h₂.

π r₁² h₁=π r₂² h₂
2STEP 2

Turn 10% more into a multiplier

"10% more" means r₂=1.1 r₁; substitute into the volume equation.

r₂=1.1 r₁, π r₁² h₁=π (1.1 r₁)² h₂
3STEP 3

Cancel the common factors

Cancel the shared π r₁² and use (1.1)²=1.21 to get h₁=1.21 h₂.

π r₁² h₁=1.21 π r₁² h₂ → h₁=1.21 h₂
4STEP 4

Read 1.21 as a percent and pick the choice

h₁=1.21 h₂ means the first height is 21% more than the second — choice (D).

h₁=1.21 h₂=h₂+0.21 h₂ → (D)
Answer
The first height is 21% more than the second.
Take r₁=10, so r₂=11, and pick h₂=100. Then V₂=π(11)²(100)=12100π. Equal volume forces V₁=π(10)² h₁=12100π, so h₁=121. The first height 121 is exactly 21% more than the second 100, confirming (D). The second 100 is 100/121≈ 82.6% of the first, not 80%, so (E) is wrong; and a 10% comparison ((A) or (B)) does not fit either, since the gap between 121 and 100 is 21%.
💡Key takeaway

Since volume stays fixed, height must fight the radius squared — a 10% wider radius opens a 21% height gap, not 10%.

  • Name the parts and set volumes equal
  • Turn 10% more into a multiplier
  • Cancel the common factors
  • Read 1.21 as a percent and pick the choice

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