Problem:
Let be a right rectangular prism (box) with edge lengths , and , together with its interior. For real , let be the set of points in 3-dimensional space that lie within a distance of some point in . The volume of can be expressed as , where , , and are positive real numbers. What is ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Split into regions:
The rectangular prism itself
The extensions of the faces of
The quarter cylinders at each edge of
The one-eighth spheres at each corner of
Region : The volume of is
Region : This volume is equal to the surface area of times (these "extensions" are just more boxes). The volume is then
Region : We see that there are quarter-cylinders, of each type. We have quarter-cylinders of height quarter-cylinders of height quarter-cylinders of height . Since quarter-cylinders make a full cylinder, the total volume is
Therefore, .
Region : There is an eighth-sphere of radius at each corner of . Since there are corners, these eighth-spheres add up to full sphere of radius . The volume of this sphere is then
Using these values:
The problems on this page are the property of the MAA's American Mathematics Competitions