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2012 AMC 10B Problem 17

Problem 17 of 25IntermediateGeometry

Jesse cuts a circular paper disk of radius 1212 along two radii to form two sectors, the smaller having a central angle of 120120 degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger?

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Solution

Each sector forms a cone with slant height 1212. The smaller sector has angle 120∘120^\circ, so its arc length is 13⋅2π⋅12=8π\dfrac13\cdot2\pi\cdot12=8\pi, giving base radius 44. Its cone height is 122−42=82\sqrt{12^2-4^2}=8\sqrt2. The larger sector has arc length 16π16\pi, giving base radius 88. Its cone height is 122−82=45\sqrt{12^2-8^2}=4\sqrt5. The volume ratio is 13π⋅42⋅8213π⋅82⋅45=1010.\dfrac{\frac13\pi\cdot4^2\cdot8\sqrt2}{\frac13\pi\cdot8^2\cdot4\sqrt5}=\dfrac{\sqrt{10}}{10}. Thus, C is the correct answer.

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Concepts: cone · sector · volume

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.