AMC 10 · 2023 · #21
Grade 10 geometry-3dPick an answer.
Shortest paths on a curved surface look hopeless until you notice what kind of curved surface this is. A cone is built entirely out of straight lines running from its tip to its rim, which means it can be slit and rolled flat without stretching anything (Visualize Spatial Relationships). Lengths along the surface survive that flattening untouched, so the curved question becomes a flat one, and on flat paper shortest means straight. Everything after that is bookkeeping about the flat picture (Draw a Diagram): where is the center, how far out are the two rims, how wide is the spread. The center is the cone's missing tip, so it has to be rebuilt from similar triangles with a named unknown (Introduce a Variable), and the two rim distances are slant heights, each its own small right-triangle computation (Identify Subproblems). Two traps sit in wait. The flattened spread is not a full turn — the rim keeps its length but lands on a bigger circle, so it wraps through a smaller angle. And the straight segment between the two points dives into the hole left by the top opening, so it is not a legal route (Eliminate Possibilities); the real shortest path is a string pulled taut around that hole (Extreme Principle), straight until it grazes the edge and then hugging it.
Flatten the surface
Flattening preserves lengths.
A cone is a rolled-up sheet of paper, so unrolling it leaves every length on the surface exactly as it was.
A cone is a rolled-up sheet, so unrolling it leaves every length on the surface exactly as it was.
▸ Why?
Unrolling moves the surface without stretching it, so distances measured along it are preserved.
▸ Why?
The rim's length is fixed, so laying it along a bigger circle makes it wrap through a smaller angle.
Rebuild the missing tip
Similarity rebuilds the tip.
Halving the radius halves everything, so the cut sits exactly halfway up and the missing tip is as tall as the shade.
10.G-SRT.B.5Introduce A VariableSlant heights become radii
The slant heights become the flat radii.
Distance from the tip survives the flattening, and on a cone that distance is the slant height.
8.G.B.7Identify SubproblemsIdentify the flat shape
The flat piece is a half-annulus.
The rim's length is fixed, so laying it along a bigger circle makes it wrap through a smaller angle.
10.G-C.B.5Visualize Spatial RelationshipsPlace the two points
Place the bug and the honey.
Flattening squeezes every angle around the shade to half its size, so opposite sides land a quarter turn apart, not half.
10.G-GPE.B.4Draw A DiagramThe straight line leaves the shade
The straight line leaves the shade.
The straight shortcut dives into the hole in the middle, which is precisely the part of the plane the lampshade does not cover.
8.G.B.8Eliminate PossibilitiesPull the string taut
Pull it taut against the inner rim.
A tight string bends only where something forces it to, so it stays straight until the hole pushes back and then hugs the edge.
10.G-C.A.2Extreme PrincipleAdd the arc and finish
Adding the arc gives six root three plus pi.
The straight tangent already eats 60° of the 90° spread, so only 30° of wrapping is left to pay for.
10.G-SRT.C.6Identify SubproblemsA cone is a rolled-up sheet of paper, so unroll it and the shortest crawl turns into a straight line — except the shade's top opening leaves a hole in that sheet, so the line has to run straight until it grazes the hole and then hug the edge the rest of the way.
- Flatten the surface into a plane
- Rebuild the cone's missing tip
- Slant heights become the two radii
- The flat piece is a half-annulus
- Place the bug and the honey
- The straight line leaves the shade
- Pull the string taut
- Add the arc and finish
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