2005 AMC 12B Problem 16

Below is the professionally curated solution for Problem 16 of the 2005 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2005 AMC 12B solutions, or check the answer key.

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Concepts:3D geometry sphere distance formula

Difficulty rating: 1660

16.

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?

$\sqrt{2}$

$\sqrt{3}$

$1 + \sqrt{2}$

$1 + \sqrt{3}$

$3$

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.

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