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2020 AMC 12B Problem 9

Problem 9 of 25EasierGeometry

A three-quarter sector of a circle of radius 44 inches together with its interior can be rolled up to form the lateral surface of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?

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Solution

The sector’s arc length is 34⋅2π⋅4=6π,\tfrac34 \cdot 2\pi \cdot 4 = 6\pi, which becomes the base circumference: 2πr=6π,2\pi r = 6\pi, so r=3.r = 3. The slant height is the sector radius 4,4, so the height is h=42−32=7.h = \sqrt{4^2 - 3^2} = \sqrt{7}. The volume is 13πr2h=13π⋅9⋅7=3π7.\frac13 \pi r^2 h = \frac13 \pi \cdot 9 \cdot \sqrt{7} = 3\pi\sqrt{7}. Thus, the correct answer is C.
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Tagged: cone · volume · Pythagorean Theorem

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