2005 AMC 12B Problem 16

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.