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 in . 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 , so .
Region 2: This volume is equal to the surface area of times (these "extensions" are just more boxes). The volume is then to get .
Region 3: We see that there are 12 quarter-cylinders, 4 of each type. We have 4 quarter-cylinders of height 1, 4 quarter-cylinders of height 3, 4 quarter-cylinders of height 4. Since 4 quarter-cylinders make a full cylinder, the total volume is . Therefore, .
Region 4: There is an eighth-sphere of radius at each corner of . Since there are 8 corners, these eighth-spheres add up to 1 full sphere of radius . The volume of this sphere is then , so .
Using these values, .
To see a diagram of , view TheBeautyofMath's explanation video (Video Solution 1).
~DrJoyo
~Edits by BakedPotato66
Video Solutions
Video Solution 1
https://youtu.be/3BvJeZU3T-M?t=1351
~IceMatrix
Video Solution 2
https://www.youtube.com/watch?v=NAZTdSecBvs ~ MathEx
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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