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 <math>3</math> by <math>4</math> by <math>5</math> units. Given that the [[volume]] of this set is <math>\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 == | ||
+ | <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 volume of the parallelepiped is <math>3 \times 4 \times 5 = 60 </math> cubic units. | ||
+ | *The volume of the external parallelepipeds is <math>2(3 \times 4 \times 1)+2(3 \times 5 \times 1 )+2(4 \times 5 \times 1)=94 </math>. | ||
+ | *There are <math>8</math> of the <math>1/8</math> spheres, each of radius <math>1</math>. Together, their volume is <math> \frac{4}{3}\pi </math>. | ||
+ | *There are <math>12</math> of the <math>1/4</math> cylinders, so <math>3</math> 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 = \boxed{505} </math>. | ||
== See also == | == See also == | ||
− | + | {{AIME box|year=2003|n=I|num-b=4|num-a=6}} | |
+ | |||
+ | [[Category:Intermediate Geometry Problems]] | ||
+ | {{MAA Notice}} |
Latest revision as of 18:58, 4 July 2013
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
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.