Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "1990 AHSME Problems/Problem 25"
(→‎Solution)
Line 12: Line 12:
 
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>.
 
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>
+
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>1\times 1\times 1</math> and its space diagonal is of length <math>\sqrt3</math>. Thus the space diagonal of the unit cube is <math>2(r\sqrt3+2r)=\sqrt3</math>. Solving for <math>r</math>, we get <math>\fbox{B}</math>
 
<asy>
 
<asy>
 
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
 
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
Line 22: Line 22:
 
label("P",(.25,.25),S);label("C",(.5,.5),N);
 
label("P",(.25,.25),S);label("C",(.5,.5),N);
 
</asy>
 
</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 10:50, 26 June 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?

$\text{(A) } 1-\frac{\sqrt{3}}{2}\quad \text{(B) } \frac{2\sqrt{3}-3}{2}\quad \text{(C) } \frac{\sqrt{2}}{6}\quad \text{(D) } \frac{1}{4}\quad \text{(E) } \frac{\sqrt{3}(2-\sqrt2)}{4}$

Solution

Let $r$ be the radius, let $C$ be the center of the cube, and let $P$ be the center of one of the eight outer spheres, noting that $PC=2r$.

Pack, in the corner of the unit cube, a smaller cube whose inner corner coincides with $P$. The cube is of dimensions $1\times 1\times 1$ and its space diagonal is of length $\sqrt3$. Thus the space diagonal of the unit cube is $2(r\sqrt3+2r)=\sqrt3$. Solving for $r$, we get $\fbox{B}$ [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

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. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=1990_AHSME_Problems/Problem_25&oldid=79089"