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Difference between revisions of "1994 AHSME Problems/Problem 11"

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==Problem==
 
==Problem==
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Three cubes of volume <math>1, 8</math> and <math>27</math> are glued together at their faces. The smallest possible surface area of the resulting configuration is
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Three cubes of volume <math>1</math>, <math>8</math> and <math>27</math> are glued together at their faces. The smallest possible surface area of the resulting configuration is
  
 
<math> \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 </math>
 
<math> \textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74 </math>
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==Solution==
 
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<asy>
 
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import three; currentprojection = orthographic(5,-30,20);
 
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triple[] P = {(0,0,0),(1,0,0),(1,1,0),(0,1,0),(0,0,1),(1,0,1),(1,1,1),(0,1,1)},P2={(1,0.5,1),(0.5,0,1),(0.5,0.5,1),(1,0,1.5),(1,0.5,1.5),(0.5,0.5,1.5),(0.5,0,1.5)},P3={(0.25,0,1),(0.25,0.25,1),(0.5,0.25,1),(0.25,0,1.25),(0.25,0.25,1.25),(0.5,0.25,1.25),(0.5,0,1.25)};
 
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void drawFrontFace(int w, int x, int y, int z){
 
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draw(P[w]--P[x] -- P[y] -- P[z] -- cycle, linewidth(0.7));
 
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/* fill(P[x] -- P[y] -- P[z] -- cycle, rgb(0.7,0.7,0.7)); */ }
 
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void drawBackFace(int w, int x, int y, int z){
 
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draw(P[w]--P[x] -- P[y] -- P[z] -- cycle, linetype("2 6")); }
 
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drawFrontFace(4,5,6,7);
 
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drawFrontFace(0,1,5,4);
 
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drawFrontFace(1,2,6,5);
 
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drawBackFace(0,1,2,3);
 
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drawBackFace(3,3,7,7);
 
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draw(P2[3]--P2[4]--P2[5]--P2[6]--cycle,linewidth(0.7));
 
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draw(P[5]--P2[3]--P2[6]--P2[1]--cycle,linewidth(0.7));
 
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draw(P[5]--P2[0]--P2[4]--P2[3]--cycle,linewidth(0.7));
 
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draw(P[5]--P2[0]--P2[2]--P2[1]--cycle,linetype("2 6"));
 
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draw(P2[2]--P2[0]--P2[4]--P2[5]--cycle,linetype("2 6"));
 
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draw(P3[0]--P2[1]--P3[2]--P3[1]--cycle,linetype("2 6"));
 
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draw(P3[2]--P3[1]--P3[4]--P3[5]--cycle,linetype("2 6"));
 
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draw(P3[0]--P2[1]--P3[6]--P3[3]--cycle,linewidth(0.7));
 
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draw(P2[1]--P3[2]--P3[5]--P3[6]--cycle,linewidth(0.7));
 
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draw(P3[3]--P3[6]--P3[5]--P3[4]--cycle,linewidth(0.7));</asy>
 
  
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We reach the minimum surface area with the configuration above. The cubes from smallest to largest have edge lengths <math>1,2,3</math> respectively. For the cube with edge <math>3</math>, the five faces that are not in contact with another cube have total surface area of <math>9\cdot 5=45</math>.  The top face of that cube has surface area <math>9-4-1=4</math>. The left face of the cube with edge length <math>2</math> has surface area <math>4-1=3</math>. The four remaining faces of that cube have total surface area <math>4\cdot 4=16</math>. The four faces of the smallest cube that are not in contact with another cube have total surface area <math>1\cdot 4=4</math>. Therefore, our total surface area is <math>45+4+3+16+4=\boxed{\textbf{(D) }72.}</math>
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==See Also==
  
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--Solution by [http://www.artofproblemsolving.com/Forum/memberlist.php?mode=viewprofile&u=200685 TheMaskedMagician]
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{{AHSME box|year=1994|num-b=10|num-a=12}}
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{{MAA Notice}}

Latest revision as of 20:50, 22 February 2024

Problem

Three cubes of volume $1$, $8$ and $27$ are glued together at their faces. The smallest possible surface area of the resulting configuration is

$\textbf{(A)}\ 36 \qquad\textbf{(B)}\ 56 \qquad\textbf{(C)}\ 70 \qquad\textbf{(D)}\ 72 \qquad\textbf{(E)}\ 74$

See Also

1994 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

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