Difference between revisions of "2003 AIME I Problems/Problem 5"
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== Solution == | == Solution == | ||
+ | <center><asy> | ||
+ | size(220); | ||
+ | import three; currentprojection = perspective(5,4,3); defaultpen(linetype("8 8")+linewidth(0.6)); | ||
+ | draw(box((0,-.1,0),(0.4,0.6,0.3))); | ||
+ | draw(box((-.1,0,0),(0.5,0.5,0.3))); | ||
+ | draw(box((0,0,-.1),(0.4,0.5,0.4))); | ||
+ | draw(box((0,0,0),(0.4,0.5,0.3)),linewidth(1.2)+linetype("1")); | ||
+ | </asy></center> | ||
+ | |||
The set can be broken into several parts: the big <math>3\times 4 \times 5</math> parallelepiped, <math>6</math> external parallelepipeds that each share a face with the large parallelepiped and have a height of <math>1</math>, the <math>1/8</math> [[sphere]]s (one centered at each [[vertex]] of the large parallelepiped), and the <math>1/4</math> [[cylinder]]s connecting each adjacent pair of spheres. | The set can be broken into several parts: the big <math>3\times 4 \times 5</math> parallelepiped, <math>6</math> external parallelepipeds that each share a face with the large parallelepiped and have a height of <math>1</math>, the <math>1/8</math> [[sphere]]s (one centered at each [[vertex]] of the large parallelepiped), and the <math>1/4</math> [[cylinder]]s connecting each adjacent pair of spheres. | ||
Revision as of 20:18, 10 June 2008
Problem
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures by
by
units. Given that the volume of this set is
where
and
are positive integers, and
and
are relatively prime, find
Solution
![[asy] size(220); import three; currentprojection = perspective(5,4,3); defaultpen(linetype("8 8")+linewidth(0.6)); draw(box((0,-.1,0),(0.4,0.6,0.3))); draw(box((-.1,0,0),(0.5,0.5,0.3))); draw(box((0,0,-.1),(0.4,0.5,0.4))); draw(box((0,0,0),(0.4,0.5,0.3)),linewidth(1.2)+linetype("1")); [/asy]](http://latex.artofproblemsolving.com/6/b/d/6bdbe041f3224fa7f40b0e2831bb24448baa3814.png)
The set can be broken into several parts: the big parallelepiped,
external parallelepipeds that each share a face with the large parallelepiped and have a height of
, the
spheres (one centered at each vertex of the large parallelepiped), and the
cylinders connecting each adjacent pair of spheres.
- The volume of the parallelepiped is
cubic units.
- The volume of the external parallelepipeds is
.
- There are
of the
spheres, each of radius
. Together, their volume is
.
- There are
of the
cylinders, so
complete cylinders can be formed. Their volumes are
,
, and
, adding up to
.
The combined volume of these parts is . Thus, the answer is
.
See also
2003 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |