{"status": "success", "data": {"description_md": "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 $h$?\n\n$$<center><img class=\"problem-image\" alt=\"[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); draw((0,0,0)--(4,0,0)--(4,4,0)--(0,4,0)--(0,0,0)^^(4,0,0)--(4,0,h)--(4,4,h)--(0,4,h)--(0,4,0)); draw(shift((1,3,1))*unitsphere,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]\" class=\"latexcenter\" height=\"252\" src=\"https://latex.artofproblemsolving.com/9/d/8/9d894dbf42883508d51bf4973c78d1368570c6fa.png\" width=\"148\"/></center>$$\n\n$\\textbf{(A) }2+2\\sqrt 7\\qquad<br>\\textbf{(B) }3+2\\sqrt 5\\qquad<br>\\textbf{(C) }4+2\\sqrt 7\\qquad<br>\\textbf{(D) }4\\sqrt 5\\qquad<br>\\textbf{(E) }4\\sqrt 7\\qquad$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>A  <span class=\"katex--inline\">4\\times 4\\times h</span>  rectangular box contains a sphere of radius  <span class=\"katex--inline\">2</span>  and eight smaller spheres of radius  <span class=\"katex--inline\">1</span> .  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  <span class=\"katex--inline\">h</span> ?</p>&#10;<p> <span class=\"katex--display\">&lt;center&gt;&lt;img class=&#34;problem-image&#34; alt=&#34;[asy] import graph3; import solids; real h=2+2*sqrt(7); currentprojection=orthographic((0.75,-5,h/2+1),target=(2,2,h/2)); currentlight=light(4,-4,4); 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,gray(0.85)); draw(shift((3,3,1))*unitsphere,gray(0.85)); draw(shift((3,1,1))*unitsphere,gray(0.85)); draw(shift((1,1,1))*unitsphere,gray(0.85)); draw(shift((2,2,h/2))*scale(2,2,2)*unitsphere,gray(0.85)); draw(shift((1,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,3,h-1))*unitsphere,gray(0.85)); draw(shift((3,1,h-1))*unitsphere,gray(0.85)); draw(shift((1,1,h-1))*unitsphere,gray(0.85)); draw((0,0,0)--(0,0,h)--(4,0,h)^^(0,0,h)--(0,4,h)); [/asy]&#34; class=&#34;latexcenter&#34; height=&#34;252&#34; src=&#34;https://latex.artofproblemsolving.com/9/d/8/9d894dbf42883508d51bf4973c78d1368570c6fa.png&#34; width=&#34;148&#34;/&gt;&lt;/center&gt;</span> </p>&#10;<p> <span class=\"katex--inline\">\\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</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "2014 AMC 12A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/14_amc12A_p18", "prev": "/problem/14_amc12A_p16"}}