AMC 10 · 2005 · #17

Grade 10 geometry-3d

Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid

volume-pyramidspatial-visualizationcoordinate-geometry spatial-visualizationcoordinate-geometryidentify-subproblems ↑ Prerequisites: volume-pyramidspatial-visualizationcoordinate-geometry

📏 Long solution 💡 3 insights 📊 Diagram

Problem

A unit cube is sliced by two planes into three prisms, and the same pair of slices is repeated a quarter turn around. The cube ends up in nine pieces. Find the volume of the piece holding one named vertex.

Pick an answer.

(A)

$\frac{1}{12}$

(B)

$\frac{1}{9}$

(C)

$\frac{1}{8}$

(D)

$\frac{1}{6}$

(E)

$\frac{1}{4}$

How to solve

Strategy Visualize Spatial Relationships

Each cut is a plane, but the picture pins it down with one straight line on a face. Putting coordinates on the cube turns every cut into a single linear equation, so the question "which of the nine pieces holds W?" becomes "which side of each plane is W on?". Once the piece is described by inequalities, slicing it at each height shows what shape it is, and one volume formula ends the problem.

1STEP 1

Put coordinates on the cube

Coordinates turn the cube into three simple ranges.

cube = {(x,y,z) : 0 ≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1}, W = (1, 0, 1)

2STEP 2

Write the Figure 1 cuts as equations

The first pair of cuts ignores one coordinate, giving two plane equations.

z = 2y and z = 2 - 2y

3STEP 3

Add the quarter-turn cuts

The quarter turn swaps the roles, so three by three gives nine pieces.

z = 2x and z = 2 - 2x

4STEP 4

Find the two inequalities W obeys

Only two inequalities actually bound the piece.

P = {(x,y,z) ∈ cube : z ＞ 2y and z ＞ 2 - 2x}

5STEP 5

Slice the piece at height z

Each cross-section is a square whose side grows linearly, so the piece is a pyramid.

slice area at height z = (z/2)² = z²/4

6STEP 6

Use the pyramid volume formula

The pyramid formula gives 1/12, choice (A).

V = 1/3 · 1/4 · 1 = 1/12

Answer

1/12

Check that the nine pieces really rebuild the cube. Each outer slab contributes an interval of length z/2 at height z, and each middle slab contributes an interval of length 1 - z. So the eight pieces that use at least one outer slab all have volume 1/12, since integrating (z/2)(z/2) or (z/2)(1-z) from 0 to 1 both give 1/12, while the centre piece has volume equal to the integral of (1-z)², namely 1/3. Then 8 times 1/12 plus 1/3 equals 2/3 + 1/3 = 1, the volume of the whole cube, so 1/12 is consistent. The size is believable too: the piece sits inside the box with 1/2 ＜ = x ＜ = 1 and 0 ＜ = y ＜ = 1/2, a box of volume 1/4, and it fills exactly one third of that box.

💡Key takeaway

Slice a solid at every height: if the slices are squares whose side grows evenly from zero, the solid is a pyramid, and a pyramid fills one third of the box around it.

Put coordinates on the cube
Write the Figure 1 cuts as equations
Add the quarter-turn cuts
Find the two inequalities W obeys
Slice the piece at height z
Use the pyramid volume formula