Problem 23 · 2004 AMC 8
Medium
Geometry & Measurement
distance-vs-time-graph
Tess runs counterclockwise around rectangular block JKLM. She lives at corner J. Which graph could represent her straight-line distance from home?
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Answer: D — Graph D.
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Hint 1 of 2
Don't compute — tell the story of the distance and match its shape. Tess leaves home (J), gets steadily farther, is farthest when she's diagonally across at L, then comes home. Where does the graph start, where does it peak, where does it end?
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Hint 2 of 2
The skill is reading a graph by its qualitative features: check three things — start value, number of peaks, end value — and eliminate. Here: starts at 0, exactly one peak (at L), ends back at 0.
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Approach: match the shape, eliminate the rest
- Trace the trip J→K→L→M→J. Distance from home starts at 0, grows as she heads to K, keeps growing to the diagonally-opposite corner L (the farthest point), then shrinks back through M to 0 at J. So the graph must start at 0, rise to a single maximum, and fall back to 0.
- Eliminate with those checkpoints: (A) only rises and never returns — out. (B) starts high and decreases — out. (C) has two peaks — out. (E) rises to a plateau and stays — out.
- Only D shows the one-hump 'leave and return' shape.
- The transferable habit: for 'which graph' problems, narrate the situation and pin down start, peaks/valleys, and end — those three features alone usually kill four of the five choices. Bonus insight: the curve is gently bowed, not made of straight segments, because straight-line distance grows like a hypotenuse — quickly at first, then leveling near the peak.
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