2005 AMC 12A Problem 17

Question

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?

\dfrac{1}{12}

\dfrac{1}{9}

\dfrac{1}{8}

\dfrac{1}{6}

\dfrac{1}{4}

The piece at

W

is a pyramid; identify its base and apex

If its base area is

B

and height is

h,

use

V = \dfrac{1}{3}Bh

The two perpendicular sets of cuts each run from top edges to midlines of the bottom face. Near

W

they carve out a square pyramid. Its base is the quarter of the bottom face adjacent to

W,

and its apex lies one unit above the center of the bottom face.

Thus its base is a square of side

\dfrac{1}{2}

and its altitude is the full height

1.

Therefore the volume is

\dfrac{1}{3} \left(\dfrac{1}{2}\right)^2 (1) = \dfrac{1}{12}.

Thus, the correct answer is A.

Problem 17 in other years