A $4\times 4\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. What is $h$? $\textbf{(A) }2+2\sqrt 7\qquad$ $ \textbf{(B) }3+2\sqrt 5\qquad$ $ \textbf{(C) }4+2\sqrt 7\qquad$ $ \textbf{(D) }4\sqrt 5\qquad$ $ \textbf{(E) }4\sqrt 7\qquad$
Solution
Center the box as $[0,4]\times[0,4]\times[0,h]$; the small spheres (radius $1$) sit at the $8$ corners, e.g. centered at $(1,1,1)$, and the large sphere (radius $2$) sits at the center $(2,2,h/2)$.
Tangency: distance from $(2,2,h/2)$ to $(1,1,1)$ equals $2+1=3$: $\sqrt{2+(h/2-1)^2}=3\Rightarrow(h/2-1)^2=7$.
A $6\times6\times h$ rectangular box contains a sphere of radius $2$ and eight smaller spheres of radius $1$. Each small sphere is tangent to three faces of the box, and the large sphere is tangent to each of the small spheres. What is $h$?
