Problem:
The region consisting of all points in three-dimensional space within units of line segment has volume . What is the length ?
Answer Choices:
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Solution:
Let . The region consists of a solid circular cylinder of radius and height , together with two solid hemispheres of radius centered at and . The volume of the cylinder is , and the two hemispheres have a combined volume of . Therefore , and .
The problems on this page are the property of the MAA's American Mathematics Competitions