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Difference between revisions of "2005 AMC 12A Problems/Problem 17"

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== Problem ==
 
== Problem ==
 
A unit [[cube]] is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex <math>W</math>?
 
A unit [[cube]] is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex <math>W</math>?
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 +
<asy>
 +
path a=(0,0)--(10,0)--(10,10)--(0,10)--cycle;
 +
path b = (0,10)--(6,16)--(16,16)--(16,6)--(10,0);
 +
path c= (10,10)--(16,16);
 +
path d= (0,0)--(3,13)--(13,13)--(10,0);
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path e= (13,13)--(16,6);
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draw(a,linewidth(0.7));
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draw(b,linewidth(0.7));
 +
draw(c,linewidth(0.7));
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draw(d,linewidth(0.7));
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draw(e,linewidth(0.7));
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draw(shift((20,0))*a,linewidth(0.7));
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draw(shift((20,0))*b,linewidth(0.7));
 +
draw(shift((20,0))*c,linewidth(0.7));
 +
draw(shift((20,0))*d,linewidth(0.7));
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draw(shift((20,0))*e,linewidth(0.7));
 +
draw((20,0)--(25,10)--(30,0),dashed);
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draw((25,10)--(31,16)--(36,6),dashed);
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draw((15,0)--(10,10),Arrow);
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draw((15.5,0)--(30,10),Arrow);
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label("$W$",(15.2,0),S);
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label("Figure 1",(5,0),S);
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label("Figure 2",(25,0),S);
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</asy>
  
 
<math>
 
<math>
 
(\mathrm {A}) \ \frac{1}{12} \qquad (\mathrm {B}) \ \frac{1}{9} \qquad (\mathrm {C})\ \frac{1}{8} \qquad (\mathrm {D}) \ \frac{1}{6} \qquad (\mathrm {E})\ \frac{1}{4}
 
(\mathrm {A}) \ \frac{1}{12} \qquad (\mathrm {B}) \ \frac{1}{9} \qquad (\mathrm {C})\ \frac{1}{8} \qquad (\mathrm {D}) \ \frac{1}{6} \qquad (\mathrm {E})\ \frac{1}{4}
 
</math>
 
</math>
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−
{{image}}
 
  
 
== Solution ==
 
== Solution ==
−
It is a [[pyramid]], so <math>\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12}</math>.
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It is a [[pyramid]] with height <math>1</math> and base area <math>\frac{1}{4}</math>, so using the formula for the volume of a pyramid, <math>\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12} \Rightarrow \boxed{(\mathrm {A})}</math>.
  
 
== See also ==
 
== See also ==
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[[Category:Introductory Geometry Problems]]
 
[[Category:Introductory Geometry Problems]]
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[[Category:3D Geometry Problems]]
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{{MAA Notice}}

Latest revision as of 12:56, 19 January 2021

Problem

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?

[asy] path a=(0,0)--(10,0)--(10,10)--(0,10)--cycle; path b = (0,10)--(6,16)--(16,16)--(16,6)--(10,0); path c= (10,10)--(16,16); path d= (0,0)--(3,13)--(13,13)--(10,0); path e= (13,13)--(16,6); draw(a,linewidth(0.7)); draw(b,linewidth(0.7)); draw(c,linewidth(0.7)); draw(d,linewidth(0.7)); draw(e,linewidth(0.7)); draw(shift((20,0))*a,linewidth(0.7)); draw(shift((20,0))*b,linewidth(0.7)); draw(shift((20,0))*c,linewidth(0.7)); draw(shift((20,0))*d,linewidth(0.7)); draw(shift((20,0))*e,linewidth(0.7)); draw((20,0)--(25,10)--(30,0),dashed); draw((25,10)--(31,16)--(36,6),dashed); draw((15,0)--(10,10),Arrow); draw((15.5,0)--(30,10),Arrow); label("$W$",(15.2,0),S); label("Figure 1",(5,0),S); label("Figure 2",(25,0),S); [/asy]

$(\mathrm {A}) \ \frac{1}{12} \qquad (\mathrm {B}) \ \frac{1}{9} \qquad (\mathrm {C})\ \frac{1}{8} \qquad (\mathrm {D}) \ \frac{1}{6} \qquad (\mathrm {E})\ \frac{1}{4}$

Solution

It is a pyramid with height $1$ and base area $\frac{1}{4}$, so using the formula for the volume of a pyramid, $\frac{1}{3} \cdot \left(\frac{1}{4}\right) \cdot (1) = \frac {1}{12} \Rightarrow \boxed{(\mathrm {A})}$.

See also

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 16 		Followed by
Problem 18
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
All AMC 12 Problems and Solutions

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