2005 AMC 12B Problem 16

Question

Eight spheres of radius

1,

one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres?

\sqrt{2}

\sqrt{3}

1 + \sqrt{2}

1 + \sqrt{3}

3

Answer: D

Concepts:3D geometry sphere distance formula

Difficulty rating: 1660

Small hint:

A unit sphere tangent to all three coordinate planes in one octant has center

(\pm1, \pm1, \pm1)

Big hint:

Add the distance from the origin to a center and one more radius

Solution:

A sphere of radius

1

tangent to the three coordinate planes in one octant has its center at a point like

(1, 1, 1),

at distance

\sqrt{1^2 + 1^2 + 1^2} = \sqrt3

from the origin.

The farthest point of that sphere from the origin is at distance

\sqrt3 + 1,

so the containing sphere has radius

1 + \sqrt3.

Thus, the correct answer is D.