AMC 8 · 2008 · #21

Grade 8 geometry-3d
Archetype Solid Volume by Decomposition └ Sub-archetype Fraction or Difference of a Bounding Solid
volume-cylinderarea-circlesspatial-visualizationline-symmetry identify-subproblems ↑ Prerequisites: volume-cylinderarea-circles
📏 Short solution 💡 2 insights 📊 Diagram
Problem
A cylinder of bologna is 6 cm long and 8 cm across (so radius 4 cm). Jerry's dashed cut is a slanted oval that passes through the cylinder's central axis, dividing it into two matching wedges. Estimate the volume of one wedge in cubic centimeters.

Pick an answer.

(A)
48
(B)
75
(C)
151
(D)
192
(E)
603

AMC 8 2008 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve
Strategy Visualize / Use Symmetry

The dashed oval looks fancy, but it is just the slanted view of a flat cut that goes straight through the cylinder's middle line (its axis). Tool #15 (Visualize / Symmetry) lets us see that this single slice splits the cylinder into two mirror-image halves of equal size. Once we believe that, the hard 3-D shape question shrinks to Tool #9 (Easier Related Problem): find the whole cylinder's volume and take half. No calculus, no fancy slicing.

1STEP 1

The 8 cm across is the diameter, so r = 4 cm, and the label along the cylinder gives h = 6 cm.

r = 4 cm, h = 6 cm
2STEP 2

The cut is a flat plane through the central axis, so reflection makes the two wedges congruent — each is exactly half the cylinder.

V_wedge = 12\frac{1}{2} V_cylinder
3STEP 3

Cylinder volume is V = π r² h = π · 16 · 6 = 96π cm³.

V_cylinder = π (4)² (6) = π · 16 · 6 = 96π cm³
4STEP 4

Halving gives the wedge: 48π ≈ 150.8 cm³, closest to choice (C).

V_wedge = 12\frac{1}{2}(96π) = 48π ≈ 48 × 3.14159 ≈ 150.8 cm³ → (C) 151
Answer
151
Sanity check the size. The whole cylinder is 96π ≈ 301.6 cm³, so half is about 150.8 cm³ — right between (B) 75 and (D) 192, and matching (C) 151 very closely. Choice (A) 48 forgets the π, (D) 192 doubles the wedge by mistake, and (E) 603 is the whole cylinder times 2. Only (C) is consistent with "half of 96π".
💡Key takeaway

When a cut goes straight through the center of a round shape, both pieces are equal — so this AMC 8 problem is really just "half of π r² h".

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