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

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.

2005 AMC 12B Problem 16

Answer: D

Difficulty rating: 1660

Solution:

Problem 16 in Other Years