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Difference between revisions of "2001 AIME II Problems/Problem 15"

 
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== Problem ==
 
== Problem ==
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Let <math>EFGH</math>, <math>EFDC</math>, and <math>EHBC</math> be three adjacent [[square]] faces of a [[cube]], for which <math>EC = 8</math>, and let <math>A</math> be the eighth [[vertex]] of the cube. Let <math>I</math>, <math>J</math>, and <math>K</math>, be the points on <math>\overline{EF}</math>, <math>\overline{EH}</math>, and <math>\overline{EC}</math>, respectively, so that <math>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <math>\overline{AE}</math>, and containing the edges, <math>\overline{IJ}</math>, <math>\overline{JK}</math>, and <math>\overline{KI}</math>. The [[surface area]] of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>, where <math>m</math>, <math>n</math>, and <math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>.
  
 
== Solution ==
 
== Solution ==
 +
<center><asy>
 +
import three; currentprojection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
 +
 +
triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8);
 +
draw((1,0,0)--(8,0,0)--(8,0,8)--(0,0,8)--(0,0,1)); draw((1,0,0)--(8,0,0)--(8,8,0)--(0,8,0)--(0,1,0),l); draw((0,8,0)--(0,8,8)); draw((0,8,8)--(0,0,8)--(0,0,1)); draw((8,8,0)--(8,8,6),l); draw((8,0,8)--(8,6,8)); draw((0,8,8)--(6,8,8));
 +
 +
draw(S--T--U--V--W--cycle); draw((0,0,1)--T--U--X--(0,2,2)--cycle); draw((0,1,0)--W--V--X--(0,2,2)--cycle);
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</asy> &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; <asy>
 +
import three; currentprojection = orthographic(camera=(1/2,1/3,-1/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2");
 +
 +
triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8);
 +
draw((1,0,0)--(8,0,0)--(8,0,8),l); draw((8,0,8)--(0,0,8)); draw((0,0,8)--(0,0,1),l); draw((8,0,0)--(8,8,0)); draw((8,8,0)--(0,8,0)); draw((0,8,0)--(0,1,0),l); draw((0,8,0)--(0,8,8)); draw((0,0,8)--(0,0,1),l); draw((8,8,0)--(8,8,6)); draw((8,0,8)--(8,6,8)); draw((0,0,8)--(0,8,8)--(6,8,8));
 +
 +
draw(S--T--U--V--W--cycle); draw((0,0,1)--T--U--X--(0,2,2)--cycle); draw((0,1,0)--W--V--X--(0,2,2)--cycle);
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</asy></center>
 +
 +
Set the coordinate system so that vertex <math>E</math>, where the drilling starts, is at <math>(8,8,8)</math>. Using a little visualization (involving some [[similar triangles]], because we have parallel lines) shows that the tunnel meets the bottom face (the xy plane one) in the line segments joining <math>(1,0,0)</math> to <math>(2,2,0)</math>, and <math>(0,1,0)</math> to <math>(2,2,0)</math>, and similarly for the other three faces meeting at the origin (by symmetry). So one face of the tunnel is the polygon with vertices (in that order), <math>S(1,0,0), T(2,0,2), U(8,6,8), V(8,8,6), W(2,2,0)</math>, and the other two faces of the tunnel are congruent to this shape.
 +
 +
Observe that this shape is made up of two congruent [[trapezoid]]s each with height <math>\sqrt {2}</math> and bases <math>7\sqrt {3}</math> and <math>6\sqrt {3}</math>. Together they make up an area of <math>\sqrt {2}(7\sqrt {3} + 6\sqrt {3}) = 13\sqrt {6}</math>. The total area of the tunnel is then <math>3\cdot13\sqrt {6} = 39\sqrt {6}</math>. Around the corner <math>E</math> we're missing an area of <math>6</math>, the same goes for the corner opposite <math>E</math> . So the outside area is <math>6\cdot 64 - 2\cdot 6 = 372</math>. Thus the the total surface area is <math>372 + 39\sqrt {6}</math>, and the answer is <math>372 + 39 + 6 = \boxed{417}</math>.
  
 
== See also ==
 
== See also ==
−
* [[2001 AIME II Problems]]
+
{{AIME box|year=2001|n=II|num-b=14|after=Last Question}}
 +
 
 +
[[Category:Intermediate Geometry Problems]]
 +
{{MAA Notice}}

Latest revision as of 14:01, 31 December 2021

Problem

Let $EFGH$, $EFDC$, and $EHBC$ be three adjacent square faces of a cube, for which $EC = 8$, and let $A$ be the eighth vertex of the cube. Let $I$, $J$, and $K$, be the points on $\overline{EF}$, $\overline{EH}$, and $\overline{EC}$, respectively, so that $EI = EJ = EK = 2$. A solid $S$ is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to $\overline{AE}$, and containing the edges, $\overline{IJ}$, $\overline{JK}$, and $\overline{KI}$. The surface area of $S$, including the walls of the tunnel, is $m + n\sqrt {p}$, where $m$, $n$, and $p$ are positive integers and $p$ is not divisible by the square of any prime. Find $m + n + p$.

Solution

[asy] import three; currentprojection = orthographic(camera=(1/4,2,3/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2"); triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8); draw((1,0,0)--(8,0,0)--(8,0,8)--(0,0,8)--(0,0,1)); draw((1,0,0)--(8,0,0)--(8,8,0)--(0,8,0)--(0,1,0),l); draw((0,8,0)--(0,8,8)); draw((0,8,8)--(0,0,8)--(0,0,1)); draw((8,8,0)--(8,8,6),l); draw((8,0,8)--(8,6,8)); draw((0,8,8)--(6,8,8)); draw(S--T--U--V--W--cycle); draw((0,0,1)--T--U--X--(0,2,2)--cycle); draw((0,1,0)--W--V--X--(0,2,2)--cycle); [/asy]             [asy] import three; currentprojection = orthographic(camera=(1/2,1/3,-1/4)); defaultpen(linewidth(0.7)); pen l = linewidth(0.5) + linetype("10 2"); triple S=(1,0,0), T=(2,0,2), U=(8,6,8), V=(8,8,6), W=(2,2,0), X=(6,8,8); draw((1,0,0)--(8,0,0)--(8,0,8),l); draw((8,0,8)--(0,0,8)); draw((0,0,8)--(0,0,1),l); draw((8,0,0)--(8,8,0)); draw((8,8,0)--(0,8,0)); draw((0,8,0)--(0,1,0),l); draw((0,8,0)--(0,8,8)); draw((0,0,8)--(0,0,1),l); draw((8,8,0)--(8,8,6)); draw((8,0,8)--(8,6,8)); draw((0,0,8)--(0,8,8)--(6,8,8)); draw(S--T--U--V--W--cycle); draw((0,0,1)--T--U--X--(0,2,2)--cycle); draw((0,1,0)--W--V--X--(0,2,2)--cycle); [/asy]

Set the coordinate system so that vertex $E$, where the drilling starts, is at $(8,8,8)$. Using a little visualization (involving some similar triangles, because we have parallel lines) shows that the tunnel meets the bottom face (the xy plane one) in the line segments joining $(1,0,0)$ to $(2,2,0)$, and $(0,1,0)$ to $(2,2,0)$, and similarly for the other three faces meeting at the origin (by symmetry). So one face of the tunnel is the polygon with vertices (in that order), $S(1,0,0), T(2,0,2), U(8,6,8), V(8,8,6), W(2,2,0)$, and the other two faces of the tunnel are congruent to this shape.

Observe that this shape is made up of two congruent trapezoids each with height $\sqrt {2}$ and bases $7\sqrt {3}$ and $6\sqrt {3}$. Together they make up an area of $\sqrt {2}(7\sqrt {3} + 6\sqrt {3}) = 13\sqrt {6}$. The total area of the tunnel is then $3\cdot13\sqrt {6} = 39\sqrt {6}$. Around the corner $E$ we're missing an area of $6$, the same goes for the corner opposite $E$ . So the outside area is $6\cdot 64 - 2\cdot 6 = 372$. Thus the the total surface area is $372 + 39\sqrt {6}$, and the answer is $372 + 39 + 6 = \boxed{417}$.

See also

2001 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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