Problem 21 · 2008 AMC 8
Medium
Geometry & Measurement
cylinder-volumepi-approximation
Jerry cuts a wedge from a 6-cm cylinder of bologna as shown by the dashed curve. Which answer choice is closest to the volume of his wedge in cubic centimeters?
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Answer: C — About 151.
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Hint 1 of 2
The dashed cut is a tilted plane that passes through the center axis — a slice through the middle splits any cylinder into two equal halves.
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Hint 2 of 2
So skip the wedge shape entirely: find the whole cylinder's volume and halve it.
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Approach: the cut through the center halves the cylinder
- The slicing plane goes through the cylinder's central axis, so the wedge is exactly half — no need to model the slanted face. Diameter 8 means radius 4, and the length is 6, so the full cylinder is π · 4² · 6 = 96π.
- Half of that is 48π. Estimate: 48π ≈ 48 · 3.14 ≈ 151.
- Sanity check: the answer choices are spread out (48, 75, 151, 192, 603), so even a rough π ≈ 3 gives 144 — clearly closest to 151, not its neighbors.
- Why this transfers: any cut passing through a solid's center of symmetry divides it into equal halves — spotting symmetry beats integrating the odd shape.
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