Difference between revisions of "2014 AMC 12A Problems/Problem 17"
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Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively. | Let <math>A</math> be the point in the same plane as the centers of the top spheres equidistant from said centers. Let <math>B</math> be the analogous point for the bottom spheres, and let <math>C</math> be the midpoint of <math>\overline{AB}</math> and the center of the large sphere. Let <math>D</math> and <math>E</math> be the points at which line <math>AB</math> intersects the top of the box and the bottom, respectively. | ||
− | Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-\sqrt2^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</math>. | + | Let <math>O</math> be the center of any of the top spheres (you choose!). We have <math>AO=1\cdot\sqrt{2}</math>, and <math>CO=3</math>, so <math>AC=\sqrt{3^2-\left(\sqrt2\right)^2}=\sqrt{7}</math>. Similarly, <math>BC=\sqrt{7}</math>. <math>\overline{AD}</math> and <math>\overline{BE}</math> are clearly equal to the radius of the small spheres, <math>1</math>. Thus the total height is <math>AD+AC+BC+BE=2+2\sqrt7</math>, or <math>\boxed{\textbf{(A)}}</math>. |
(Solution by AwesomeToad) | (Solution by AwesomeToad) |
Revision as of 13:10, 4 August 2021
Problem
A rectangular box contains a sphere of radius
and eight smaller spheres of radius
. 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
?
![[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]](http://latex.artofproblemsolving.com/9/d/8/9d894dbf42883508d51bf4973c78d1368570c6fa.png)
Solution 1
Let be the point in the same plane as the centers of the top spheres equidistant from said centers. Let
be the analogous point for the bottom spheres, and let
be the midpoint of
and the center of the large sphere. Let
and
be the points at which line
intersects the top of the box and the bottom, respectively.
Let be the center of any of the top spheres (you choose!). We have
, and
, so
. Similarly,
.
and
are clearly equal to the radius of the small spheres,
. Thus the total height is
, or
.
(Solution by AwesomeToad)
Solution 2
Let be the center of the large sphere and
be the center of any small sphere. Let
be a vertex of the rectangular prism closest to point
. Let
be the point on the edge of the prism such that
and
are perpendicular. Let points
and point
lie on
and
respectively such that
and
are perpendicular at
.
is the radii of the spheres, so
.
is the shortest length between the center of a small sphere and the edge of the prism, so
. Similarly,
. Since
is a rectangle,
. Since
,
. Then,
.
is the length from
to the top of the prism or
. Thus,
. The prism is symmetrical, so
(Solution by BJHHar)
Solution 3
take a cross section and see that h is made up of two radii of the circle plus some radical expression. the only choice satisfying this condition is (a)
See Also
2014 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.