AMC 10 · 2014 · #23

Grade 8 geometry-3d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Figure Constraint Equations for Unknown Quantity
volume-conevolume-spheresimilar-trianglespythagorean-theoremquadratic-equations convert-to-algebra ↑ Prerequisites: volume-spheresimilar-triangles
📏 Long solution 💡 4 insights 📊 Diagram
Problem
A sphere sits inside a truncated right circular cone (a frustum), touching the top disk, the bottom disk, and the slanted side all at once. The frustum's volume is exactly twice the sphere's volume. Find the ratio of the bottom base radius to the top base radius.

Pick an answer.

(A)
$\dfrac32$
(B)
$\dfrac{1+\sqrt5}2$
(C)
$\sqrt3$
(D)
2
(E)
$\dfrac{3+\sqrt5}2$

AMC 10 2014 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize Spatial Relationships

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.

1STEP 1

Slice through the axis

Slice through the axis: the sphere is a circle of radius ρ inscribed in a trapezoid with bases 2R and 2r, so h = 2ρ.

h = 2ρ
2STEP 2

Slant leg from tangent lengths

Equal tangents make the slant leg R + r, so (R+r)² = (R-r)² + h² gives h² = 4Rr and hence ρ = √(Rr).

(R+r)² = (R-r)² + h² → h² = 4Rr → ρ = √(Rr)
3STEP 3

Write both volumes

Sphere: 43\frac{4}{3}π ρ³ = 43\frac{4}{3}π Rr√(Rr). Frustum, with h = 2√(Rr): 23\frac{2}{3}π √(Rr) (R² + Rr + r²).

V_sph = 4/3π (Rr)³/2, V_fr = 2π√(Rr)/3(R² + Rr + r²)
4STEP 4

Apply the volume condition

Set frustum = 2 × sphere; the shared factor 23\frac{2}{3}π√(Rr) cancels, leaving R² + Rr + r² = 4Rr, i.e. R² - 3Rr + r² = 0.

R² + Rr + r² = 4Rr → R² - 3Rr + r² = 0
5STEP 5

Solve for the ratio

Divide by r² and let x = R/r: x² - 3x + 1 = 0, and R ＞ r keeps the larger root (3+√5)/2 ≈ 2.62 — choice (E).

x² - 3x + 1 = 0 → x = (3+√(5))/2 → (E)
Answer
(3+√5)/2
Plug x = (3+√(5))/2 back in: x² - 3x + 1 = 0 holds exactly, and numerically x ≈ 2.618, comfortably bigger than 1 as required for a bottom base wider than the top. Sanity on the volumes: with r = 1, R = 2.618, the height is h = 2√(Rr) ≈ 3.24, giving frustum volume ≈ 12.13 and sphere volume ≈ 6.06 — a ratio of 2.00, matching the condition. The smaller root (3-√(5))/2 ≈ 0.38 is just the reciprocal (the same shape viewed upside down), correctly rejected because R ＞ r. Answer (E) stands.
💡Key takeaway

Cut 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

More like this

Same archetype — closest grade level first.