2014 AMC 12A Problems/Problem 17

Problem

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 $h$ ?

$[asy] import three; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(1,0,3); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,black); draw(shift((3,3,1))*unitsphere,black); draw(shift((3,1,1))*unitsphere,black); draw(shift((1,1,1))*unitsphere,black); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,black); draw(shift((1,3,h-1))*unitsphere,black); draw(shift((3,3,h-1))*unitsphere,black); draw(shift((3,1,h-1))*unitsphere,black); draw(shift((1,1,h-1))*unitsphere,black); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]$

$\textbf{(A) }2+2\sqrt 7\qquad \textbf{(B) }3+2\sqrt 5\qquad \textbf{(C) }4+2\sqrt 7\qquad \textbf{(D) }4\sqrt 5\qquad \textbf{(E) }4\sqrt 7\qquad$

Page

Toolbox

Search

2014 AMC 12A Problems/Problem 17

Problem