Difference between revisions of "1990 AHSME Problems/Problem 25"
(Created page with "== Problem == Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to t...") |
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\text{(C) } \frac{\sqrt{2}}{6}\quad | \text{(C) } \frac{\sqrt{2}}{6}\quad | ||
\text{(D) } \frac{1}{4}\quad | \text{(D) } \frac{1}{4}\quad | ||
| − | \text{(E) } \frac{\sqrt{3}(2-\ | + | \text{(E) } \frac{\sqrt{3}(2-\sqrt2)}{4}</math> |
== Solution == | == Solution == | ||
| − | <math> | + | Let <math>r</math> be the radius, let <math>C</math> be the center of the cube, and let <math>P</math> be the center of one of the eight outer spheres, noting that <math>PC=2r</math>. |
| + | Pack, in the corner of the unit cube, a smaller cube whose inner corner coincides with <math>P</math>. The cube is of dimensions <math>r\times r\times r</math> and its space diagonal is of length <math>r\sqrt3</math>. Thus the space diagonal of the unit cube is <math>2(r\sqrt3+PC)=2r(\sqrt3+2)</math>. But this must equal <math>\sqrt3</math>. Solving for <math>r</math>, we get <math>\fbox{B}</math> | ||
| + | <asy> | ||
| + | draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); | ||
| + | for(int i=0;i<4;++i)draw(rotate(i*90,(.5,.5))*circle((0.25,0.25),0.225)); | ||
| + | //draw(circle((0.75,0.25),0.225)); | ||
| + | draw(circle((0.5,0.5),0.19),dashed); | ||
| + | draw((0,0)--(.5,.5),dotted); | ||
| + | dot((.25,.25));dot((.5,.5)); | ||
| + | label("P",(.25,.25),S);label("C",(.5,.5),N); | ||
| + | </asy> | ||
== See also == | == See also == | ||
{{AHSME box|year=1990|num-b=24|num-a=26}} | {{AHSME box|year=1990|num-b=24|num-a=26}} | ||
Revision as of 14:32, 4 February 2016
Problem
Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere?
Solution
Let 
be the radius, let 
be the center of the cube, and let 
be the center of one of the eight outer spheres, noting that 
.
Pack, in the corner of the unit cube, a smaller cube whose inner corner coincides with . The cube is of dimensions 
and its space diagonal is of length 
. Thus the space diagonal of the unit cube is 
. But this must equal 
. Solving for , we get 
![[asy] draw((0,0)--(0,1)--(1,1)--(1,0)--cycle); for(int i=0;i<4;++i)draw(rotate(i*90,(.5,.5))*circle((0.25,0.25),0.225)); //draw(circle((0.75,0.25),0.225)); draw(circle((0.5,0.5),0.19),dashed); draw((0,0)--(.5,.5),dotted); dot((.25,.25));dot((.5,.5)); label("P",(.25,.25),S);label("C",(.5,.5),N); [/asy]](http://latex.artofproblemsolving.com/5/4/3/543f1a29214ff2fcfcd22b6ddf7593549cda1c83.png)
See also
| 1990 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 24 |
Followed by Problem 26 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
