Problem 22 · 1997 AJHSME
Stretch
Geometry & Measurement
volume-scaling
A two-inch cube (2 × 2 × 2) of silver weighs 3 pounds and is worth $200. How much is a three-inch cube of silver worth?
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Answer: E — $675.
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Hint 1 of 2
Silver is worth its weight, and weight comes from how much METAL is there — that's volume, not side length. Going from a 2-inch to a 3-inch cube isn't 1.5× the value.
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Hint 2 of 2
Value tracks volume. When a 3-D solid scales, volume grows with the CUBE of the side ratio, so reason in unit cubes.
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Approach: value is proportional to volume
- Slice the 2-inch cube into unit cubes: 2³ = 8 of them, sharing $200, so each unit cube is worth $200 ÷ 8 = $25.
- A 3-inch cube is 3³ = 27 unit cubes, worth 27 × $25 = $675.
- Trap: the side only grows from 2 to 3 (×1.5), but value grows ×(3/2)³ = ×3.375, giving 200 × 3.375 = $675. The weight (3 lb) is a decoy — you never needed it.
- You'll see it again: scale a length by k and area scales by k², volume (and anything proportional to it, like mass or value) by k³.
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