AMC 10 · 2011 · #15
Grade 10 geometry-3dPick an answer.
Tool #17 (Visualize Spatial Relationships): the whole problem collapses once you cut the solid with the right vertical plane — one chosen so that a triangular face stands square to it and therefore shows up as a single line. Tool #4 (Introduce a Variable): call half the base edge m; then every length in the slice is a formula in m and the tangency condition becomes one equation. Tool #7 (Identify Subproblems): the job splits into three separate questions — where is the hemisphere centred, what does tangency say numerically, and does the touch really land on the dome. Tool #3 (Eliminate Possibilities): the last step must show no other base edge can work, which needs the distance to move in one direction only as m grows.
Symmetry pins the hemisphere's centre
Symmetry puts the centre at the middle.
Being the same distance from all four slanted walls leaves exactly one spot on the floor — the middle.
10.G-CO.A.3Visualize Spatial RelationshipsName the half-edge and cut one vertical slice
One vertical slice holds the whole problem.
A wall that stands square to the cutting plane leaves only its edge line behind, so a distance in space becomes a distance on paper.
10.G-MG.A.1Introduce A VariableTangency means a perpendicular of length 2
Touching means a perpendicular of the radius.
A surface that just grazes a ball meets it head-on, so the radius to the touching point is the shortest line from the centre to that surface.
10.G-C.A.2Identify SubproblemsSimilar right triangles give that distance
Similar right triangles give that distance.
The altitude to the hypotenuse cuts a right triangle into two smaller copies of itself, so the short-to-long ratio is the same in all three.
The altitude to the hypotenuse cuts a right triangle into two smaller copies of itself.
▸ Why?
Each piece shares an angle with the whole and has its own right angle, so all three have the same shape.
▸ Why?
The tangency gives that right angle for free, since a surface grazing a ball meets its radius square on.
Solve for the half-edge
Solving gives the half-edge, so the edge is 3√2.
Clearing the square root turns the tangency condition into an ordinary quadratic in one unknown.
8.EE.A.2Introduce A VariableCheck the touch lands on the dome, not below the floor
The touch point really lands on the dome.
Half a ball only counts as touching if the touching point is on the half you actually kept.
8.G.B.7Visualize Spatial RelationshipsNo other base edge can work
The distance grows steadily, so no other edge works.
If the distance only ever grows with the base, one base size hits the radius exactly and every other size either misses or cuts through.
10.G-GPE.B.4Eliminate PossibilitiesTangent means the distance from the centre equals the radius, so cut the solid with a plane that stands square to one face and the whole three-dimensional condition becomes a single altitude in a right triangle.
- Symmetry pins the hemisphere's centre
- Name the half-edge and cut one vertical slice
- Tangency means a perpendicular of length 2
- Similar right triangles give that distance
- Solve for the half-edge
- Check the touch lands on the dome, not below the floor
- No other base edge can work
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