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 $W?$

$\dfrac{1}{12}$

$\dfrac{1}{9}$

$\dfrac{1}{8}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

Solution:

The two sets of cuts each run from a top edge down to the midline of the bottom face. Near $W$ they carve out a pyramid whose apex is the top vertex directly above $W.$

Its base is a square of side $\dfrac{1}{2}$ (a quarter of the bottom face) 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.

2005 AMC 12A Problem 17

Solution:

Problem 17 in Other Years