AMC 10 · 2014 · #19
Grade 8 geometry-3d
Pick an answer.
A sphere inside a cone is hard to reason about in 3D, so the load-bearing move is Tool #17 (Visualize Spatial Relationships): slice the solid with the vertical plane through its axis. The sphere becomes a circle inscribed in an isosceles trapezoid — a flat picture (Tool #1) where tangency turns into two facts we can measure: the height equals the sphere's diameter, and the slant leg equals the sum of the two base radii. Name the three radii (Tool #4), then Tool #13 (Convert to Algebra) turns 'frustum volume = twice sphere volume' into a single equation in the ratio R/r, which factors to a quadratic.
Slice through the axis
The two flat faces make the height a diameter.
A round ball wedged between two parallel plates just fits — the gap between the plates is exactly one ball-diameter.
7.G.A.3Visualize Spatial RelationshipsSlant leg from tangent lengths
Tangent lengths give the slanted side.
Two tangent lines drawn from the same point to a circle always have equal reach, so the sloped edge is just the two base radii laid end to end.
Two tangent lines drawn from the same point reach the circle equally far, so the slant edge is the two radii added.
▸ Why?
A tangent meets the radius at its touch point square on, making each tangent a leg of a right triangle.
▸ Why?
Those right triangles share a hypotenuse and a radius, so their remaining legs are forced to match.
Write both volumes
Both volumes write out in the same letters.
The frustum volume is the average-ish cross-section rule R²+Rr+r², and the sphere volume is the familiar four-thirds-pi-r-cubed.
8.G.C.9Introduce A VariableApply the volume condition
The volume condition simplifies beautifully.
Both sides share the same messy factor √(Rr), so cancelling it strips the problem down to a clean relation between R and r.
8.EE.C.7Convert To AlgebraSolve for the ratio
Its positive root gives choice (E).
The whole equation only depends on the ratio R/r, so calling it x collapses two unknowns into one quadratic with a clean radical answer.
8.EE.A.2Introduce A VariableCut the ball-in-a-cone straight down the middle: the ball becomes a circle inside a trapezoid, the height equals the ball's diameter, and the slanted side equals the two radii added together. Those facts turn 'volume is double' into x² - 3x + 1 = 0, whose big root (3+√(5))/2 is the answer.
- Slice through the axis
- Slant leg from tangent lengths
- Write both volumes
- Apply the volume condition
- Solve for the ratio