Difference between revisions of "2003 AIME I Problems/Problem 5"
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== Problem == | == Problem == | ||
| − | Consider the set of | + | Consider the [[set]] of [[point]]s that are inside or within one unit of a [[rectangular prism|rectangular parallelepiped]] (box) that measures 3 by 4 by 5 units. Given that the [[volume]] of this set is <math>\displaystyle \frac{m + n\pi}{p}, </math> where <math> m, n, </math> and <math> p </math> are [[positive]] [[integer]]s, and <math> n </math> and <math> p </math> are [[relatively prime]], find <math> m + n + p. </math> |
== Solution == | == Solution == | ||
| − | The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds, the 1/ | + | {{image}} |
| + | The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds that each share a [[face]] with the large parallelepiped, the <math>\frac{1}{8}</math>th [[sphere]]s (one centered at each [[vertex]] of the large parallelepiped), and the <math>\frac{1}{4}</math>th [[cylinder]]s connecting each adjacent pair of spheres. | ||
| + | *The volume of the parallelepiped is <math>3 \cdot 4 \cdot 5 = 60 </math> cubic units. | ||
| + | *The volume of the external parallelepipeds is <math>\displaystyle 2(3 \cdot 4)+2(3 \cdot 5)+2(4 \cdot 5)=94 </math>. | ||
| + | *There are 8 <math>\frac{1}{8}</math>th spheres, each of radius <math>1</math>. Together, their volume is <math> \frac{4}{3}\pi </math>. | ||
| + | *There are 12 <math>\frac{1}{4}</math>th cylinders, so 3 complete cylinders can be formed. Their volumes are <math> 3\pi </math>, <math> 4\pi </math>, and <math> 5\pi </math>, adding up to <math>12\pi</math>. | ||
| − | + | The combined volume of these parts is <math> 60+94+\frac{4}{3}\pi+12\pi = \frac{462+40\pi}{3} </math>. Thus, the answer is <math> m+n+p = 462+40+3 = 505 </math>. | |
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| − | The combined volume of these parts is <math> 60+94+\ | ||
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| − | <math> m+n+p = 462+40+3 = 505 </math> | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=2003|n=I|num-b=4|num-a=6}} | |
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[[Category:Intermediate Geometry Problems]] | [[Category:Intermediate Geometry Problems]] | ||
Revision as of 16:57, 8 March 2007
Problem
Consider the set of points that are inside or within one unit of a rectangular parallelepiped (box) that measures 3 by 4 by 5 units. Given that the volume of this set is 
where 
and 
are positive integers, and 
and are relatively prime, find 
Solution
An image is supposed to go here. You can help us out by creating one and editing it in. Thanks.
The set can be broken into several parts: the big parallelepiped, the 6 external parallelepipeds that each share a face with the large parallelepiped, the 
th spheres (one centered at each vertex of the large parallelepiped), and the 
th cylinders connecting each adjacent pair of spheres.
- The volume of the parallelepiped is
cubic units. - The volume of the external parallelepipeds is
. - There are 8
th spheres, each of radius 
. Together, their volume is 
. - There are 12
th cylinders, so 3 complete cylinders can be formed. Their volumes are 
, 
, and 
, adding up to 
.
cubic units.
.
. Together, their volume is 
.
, 
, 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 | ||
