AMC 10 · 2008 · #11

Grade 8 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid
volume-conesimilar-figuresdilation spatial-visualizationwork-backwards ↑ Prerequisites: volume-coneexponents
📏 Medium solution 💡 2 insights
Problem
A cone stands on its base and a horizontal plane cuts it, leaving one eighth of its volume above the cut. Find how far the cut sits above the base.

Pick an answer.

(A)
4000
(B)
$2000(4-\sqrt{2})$
(C)
6000
(D)
6400
(E)
7000
How to solve
Strategy Visualize Spatial Relationships

Everything here rests on one spatial fact: a level cut across a cone leaves a smaller cone on top that is a shrunken copy of the whole thing. That fact is what makes the problem solvable at all, because the base radius is never given and only cancels if the top piece is genuinely similar to the whole. So the first job is to check that the cut really is parallel to the base, not to start computing. After that, a vertical cross-section turns the solid into a triangle where the shrink factor is visible, a named variable carries the shrink into the volume formula, and working backwards from the volume fraction recovers the height. The last move is the one that is easiest to skip: the height recovered is the height of the part above water, and the question asks for the depth below it.

1STEP 1

Check the cut is parallel to the base

A parallel cut leaves a smaller cone with the same apex.

water plane ∥ base plane ⟹ above-water piece is a cone with the same apex
2STEP 2

Slice down the axis to see the shrink

Every one of its lengths shrinks by the same factor.

k = h/8000, waterline radius = kR, above-water height = h = 8000k
3STEP 3

Turn the shrink into a volume ratio

Volume therefore shrinks by that factor cubed.

V_above = 1/3π (kR)² (8000k) = k³ · 1/3π R² (8000) = k³ V
4STEP 4

Work backwards to the shrink factor

A cube root turns one eighth into one half.

k³ = 1/8 ⟹ k = ∛(1/8) = 1/2 ⟹ h = 8000 · 1/2 = 4000
5STEP 5

Turn the height above into the depth below

Subtracting from the full height gives 4000.

d = 8000 - h = 8000 - 4000 = 4000
6STEP 6

Check it works, and that nothing else does

The relation is strictly decreasing, so it is the only answer, choice (C).

(1/3π (R/2)² (4000))/(1/3π R² (8000)) = 1000/8000 = 1/8; d ↦ (1-d/8000)³ is strictly decreasing
Answer
4000
The wrong choices are exactly the wrong exponents, which is a good sign the intended trap is the cube. Treating the volume fraction as if it were the height fraction gives a dry height of 8000 · 1/8 = 1000 and a depth of 7000, which is choice (E). Treating volume as scaling with the square instead of the cube gives k² = 1/8, so k = 1/2√(2), a dry height of 2000√(2), and a depth of 8000 - 2000√(2) = 2000(4-√(2)), which is choice (B) exactly. Choice (B) being the square-law answer is strong evidence that the cube law is what the problem is testing, and the cube law is the one derived here from the volume formula rather than assumed. The size of 4000 also passes a common-sense check: a mountain is much wider near the bottom than near the top, so the top eighth of the volume should be a tall slice, taller than the eighth of the height a naive reading suggests, and 4000 is indeed far more than 1000.
💡Key takeaway

Cutting a cone level partway down leaves a smaller copy of the same cone, and shrinking every direction by half shrinks the volume to an eighth.

  • Check the cut is parallel to the base
  • Slice down the axis to see the shrink
  • Turn the shrink into a volume ratio
  • Work backwards to the shrink factor
  • Turn the height above into the depth below
  • Check it works, and that nothing else does

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