AMC 10 · 2008 · #8

Grade 8 geometry-3d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Multiplicative Rate or Series
volume-rectangular-prismsurface-areaexponents convert-to-algebraidentify-subproblems ↑ Prerequisites: volume-rectangular-prismsurface-area
📏 Medium solution 💡 1 insight
Problem
One cube has volume one, and a second cube has exactly twice its surface area. Find the second cube's volume.

Pick an answer.

(A)
$\sqrt{2}$
(B)
2
(C)
$2\sqrt{2}$
(D)
4
(E)
8
How to solve
Strategy Introduce a Variable

Volume and surface area are never compared directly; each one is a formula in the side length. Tool #4 (Introduce a Variable) names that side length, which is the single quantity both facts talk about. Tool #7 (Identify Subproblems) splits the work into a short chain: volume 1 gives the first side, the first side gives its surface area, doubling gives the target surface area, the target gives the second side, and the second side gives the volume. Because the choices are five specific numbers, Tool #3 (Eliminate Possibilities) supplies a check that runs backwards — test each candidate volume against the surface-area condition — which confirms that exactly one number can be right, not just that the chain produced a number.

1STEP 1

Turn volume 1 into a surface area

The first cube's volume gives its surface area.

s³ = 1 → s = 1, S = 6 · 1² = 6, 2S = 12
2STEP 2

Solve for the second side length

Doubling and solving gives the second side.

6t² = 12 → t² = 2 → t = √(2)
3STEP 3

Cube the side to get the volume

Cubing it gives 2√(2).

V = (√(2))³ = (√(2))² · √(2) = 2√(2)
4STEP 4

Check every choice against the condition

A direct relation between area and volume confirms 2√(2), choice (C).

S³ = 216V², 12³ = 1728 = 216V² → V² = 8 → V = 2√(2) (C)
Answer
2√(2)
The answer 2√(2) ≈ 2.83 sits between 2 and 4, which fits: the side grew by a factor of √(2) ≈ 1.41, and 1.41³ ≈ 2.83. The wrong choices are exactly the tempting mismatches. Choice (B) 2 assumes volume doubles along with surface area, but a cube of volume 2 has side ∛(2) and surface area 6∛(4) ≈ 9.5, not 12. Choice (E) 8 comes from doubling the side, which multiplies surface area by 4 (6 · 2² = 24), far past twice. Choice (D) 4 gives surface area about 15.1 and choice (A) √(2) about 7.6. Since 216V² is strictly increasing for V ＞ 0, at most one volume can meet the condition, and 2√(2) meets it exactly — so (C) is not merely the best of five, it is the only cube that works.
💡Key takeaway

Surface area follows the side squared and volume follows the side cubed, so doubling the surface area stretches the side by only √(2) and the volume by (√(2))³ = 2√(2).

  • Turn volume 1 into a surface area
  • Solve for the second side length
  • Cube the side to get the volume
  • Check every choice against the condition

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