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?
The volume of a cone is 31πr2h, 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.