AMC 10 · 2002 · #19
Grade 7 geometry-2dPick an answer.
AMC 10 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
The reachable region is an odd shape, so Tool #7 (Identify Subproblems) breaks it into pieces that are each a plain circular sector whose area is easy. Tool #1 (Draw a Diagram) pins down the angles: the doghouse blocks part of the swing at the tie-corner, and the rope can bend around the two neighboring corners. Tool #17 (Visualize Spatial Relationships) tracks how much rope is left after it wraps a corner and how far it can then swing. Add the sectors to get the total.
Swing the rope around the tie-corner
At the tie-corner the doghouse takes up the 120° interior angle, so the rope swings the leftover 240° at its full 2-yard length.
The wall takes up the 120° corner, so the rope is free to sweep the leftover 240° around it.
4.MD.C.7Draw A DiagramArea of the big sector
A full radius-2 circle has area 4π, and 240° is 2/3 of it, so the main piece is 8π/3.
Taking a fraction of a circle's area is just that fraction times π r².
Taking a fraction of a circle's area is just that fraction times pi times the radius squared.
▸ Why?
A sector is the share of the whole circle its angle takes, so the angle names the fraction directly.
▸ Why?
The whole circle's area is pi times the radius squared, so that fraction of it needs no further work.
Rope wraps around each neighbor corner
One side eats 1 yard, leaving 1, which bends 60° past that corner before the next wall stops it — and the mirror corner matches.
Past the corner the rope is shorter and only a small wedge opens up before the next wall blocks it.
4.MD.C.7Visualize Spatial RelationshipsArea of the two small sectors
Each wedge is 1/6 of a unit circle, so π/6 apiece, and the pair adds π/3.
Two matching 60° slivers of a unit circle add up to a 120° slice, which is 1/3 of the circle.
7.G.B.4Identify SubproblemsAdd every piece
The pieces do not overlap, so 8π/3 + π/3 = 3π is everything the dog can reach, choice (E).
The regions do not overlap, so the total reachable area is simply their areas added.
7.G.B.6Identify SubproblemsWhen a rope longer than one wall bends around a corner, break the space it covers into circle slices — a big slice at the tie point plus small ones where the rope wraps — and add them up.
- Swing the rope around the tie-corner
- Area of the big sector
- Rope wraps around each neighbor corner
- Area of the two small sectors
- Add every piece
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