2005 AMC 12A Problems/Problem 17

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 $W$ ?

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

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

Solution

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

2005 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2005 AMC 12A Problems/Problem 17

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