2020 AMC 10B Problems/Problem 20
Contents
Problem
Let be a right rectangular prism (box) with edges lengths
and
, together with its interior. For real
, let
be the set of points in
-dimensional space that lie within a distance
of some point
. The volume of
can be expressed as
, where
and
are positive real numbers. What is
Solution
Split into 4 regions:
1. The rectangular prism itself
2. The extensions of the faces of
3. The quarter cylinders at each edge of
4. The one-eighth spheres at each corner of
Region 1: The volume of is 12, so
Region 2: The volume is equal to the surface area of times
. The surface area can easily be computed to be
, so
.
Region 3: The volume of each quarter cylinder is equal to . The sum of all such cylinders must equal
times the sum of the edge lengths. This can easily be computed as
, so the sum of the volumes of the quarter cylinders is
, so
Region 4: There is an eighth of a sphere of radius at each corner. Since there are 8 corners, these add up to one full sphere of radius
. The volume of this sphere is
, so
.
Using these values,
~DrJoyo
Video Solution
~IceMatrix
See Also
2020 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 19 |
Followed by Problem 21 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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