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From a circular piece of paper with radius $BC$, Jeff removes the unshaded sector shown. Using the larger shaded sector, he joins edge $BC$ to edge $BA$ (without overlap) to form a cone of radius $12$ centimeters and of volume $432\pi$ cubic centimeters. What is the number of degrees in the measure of angle $ABC$ of the sector that is not used?

 Jul 7, 2023
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The volume of a cone is 31​πr2h, where r is the radius of the base and h is the height of the cone. Since the cone has a volume of 432π cubic centimeters and a radius of 12 centimeters, its height is h=12π432π​=36 centimeters.

The cone is formed by joining the two bases of the sector, which have radius 12 and AB=BC=12. Therefore, the central angle of the sector that is not used is 12360∘​=30∘.

The unshaded sector has central angle 30∘, so the shaded sector has central angle 360∘−30∘=330∘.

The answer is 330​ degrees.

 Jul 7, 2023

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