Problem 18 · 2008 AMC 8
Medium
Geometry & Measurement
arc-lengthpath-decomposition
Two circles that share the same center have radii 10 meters and 20 meters. An aardvark runs along the path shown, starting at A and ending at K. How many meters does the aardvark run?
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Answer: E — 20π + 40.
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Hint 1 of 2
A curvy path looks scary, but it's only two kinds of pieces — circular arcs and straight segments. Sort every piece into one bin or the other and the π's stay separate from the plain numbers.
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Hint 2 of 2
For an arc, ask "what fraction of the full circle is it?" then take that fraction of the circumference 2πr; each curved corner here is a quarter-turn.
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Approach: split the path into arcs (the π part) and straight segments (the plain part)
- Arcs first: the path curves a quarter-way around the big circle (radius 20) — that's ¼ · 2π · 20 = 10π — and a quarter-way around the small circle (radius 10) twice, each ¼ · 2π · 10 = 5π. Arc total: 10π + 5π + 5π = 20π.
- Now the straight pieces: two radial hops crossing the ring (each 20 − 10 = 10) plus one straight run across the small circle's diameter (2 · 10 = 20). Straight total: 10 + 10 + 20 = 40.
- Add the bins: 20π + 40. Keeping π-terms apart from whole numbers means there's nothing to combine — just stack them.
- Why this transfers: any "length of a curvy track" problem yields to the same split — arcs become (fraction)×2πr, straights are ordinary distances, and the answer is a tidy (multiple of π) + (whole number).
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