AMC 10 · 2005 · #16

Grade 10 geometry-3d
Archetype Coordinate Plane Figure Computation └ Sub-archetype Pythagorean / Distance Formula for a Length
space-diagonal-formulacoordinate-geometryoptimization spatial-visualizationeasier-related-problemextreme-principle ↑ Prerequisites: pythagorean-theoremcoordinate-geometryspace-diagonal-formula
📏 Long solution 💡 3 insights
Problem
Eight unit balls each sit in one octant, touching all three coordinate planes. Find the radius of the smallest ball centred at the origin that contains them all.

Pick an answer.

(A)
$\sqrt{2}$
(B)
$\sqrt{3}$
(C)
$1+\sqrt{2}$
(D)
$1+\sqrt{3}$
(E)
3
How to solve
Strategy Visualize Spatial Relationships

Three dimensions are hard to picture all at once, so first pin the eight centers with coordinates, then run the same problem in two dimensions where it can be drawn on paper. The word smallest is a signal to hunt for the extreme point: the single point of the eight small spheres that is farthest from the origin. Find that distance and the answer is forced, because R must be at least that and that value already works.

1STEP 1

Find the eight centers

Tangency fixes every coordinate, so the centres are cube vertices.

centers=(± 1,± 1,± 1)
2STEP 2

Rehearse it in two dimensions

The flat version rehearses the whole argument with one fewer coordinate.

√(1²+1²)+1=√(2)+1 (2D warm-up)
3STEP 3

Distance from origin to a center

Two uses of the Pythagorean theorem give the centre distance √(3).

|(± 1,± 1,± 1)|=√(1²+1²+1²)=√(3)≈ 1.732
4STEP 4

Reach for the farthest point

The farthest point lies one radius beyond the centre, and that bound is reached.

OP ≤ OC+CP=√(3)+1, with equality at P=(1+1/√(3))C
5STEP 5

Match to a choice

So the smallest radius is 1+√(3), choice (D).

R=1+√(3)≈ 2.732
Answer
1+√(3)
A rough bound settles it before any exact work. Each center is more than 1.7 from the origin and each small sphere sticks out one more unit, so R must be more than 2.7. That alone kills √(2)≈ 1.41, √(3)≈ 1.73 and 1+√(2)≈ 2.41. Between the two survivors, 3 is strictly larger than 1+√(3)≈ 2.73, and a radius of 1+√(3) already works, so 3 is not the smallest. Sanity check the 2D rehearsal too: it gave 1+√(2), which is exactly choice (C) - the trap for anyone who solves the flat version by accident.
💡Key takeaway

To enclose a ball, aim past its center: the farthest point is always one radius beyond the center, straight along the line from where you are standing.

  • Find the eight centers
  • Rehearse it in two dimensions
  • Distance from origin to a center
  • Reach for the farthest point
  • Match to a choice

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