Difference between revisions of "2020 AMC 10B Problems/Problem 20"
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Region 2: The volume is equal to the surface area of B times r. The surface area can easily be computed to be 38, so <math>c=38</math>. | Region 2: The volume is equal to the surface area of B times r. The surface area can easily be computed to be 38, so <math>c=38</math>. | ||
| − | Region 3: The volume of each quarter cylinder is equal to | + | Region 3: The volume of each quarter cylinder is equal to <math>(\pi*r^2*h)/4</math> |
==Video Solution== | ==Video Solution== | ||
Revision as of 02:55, 8 February 2020
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 the volume into 4 regions:
1. The rectangular prism itself, 2. the extensions of the faces of B, 3. the quarter cylinders at each edge of B, 4. the one-eighth spheres at each corner of B.
Region 1: The volume of B is 12, so 
Region 2: The volume is equal to the surface area of B times r. The surface area can easily be computed to be 38, so 
.
Region 3: The volume of each quarter cylinder is equal to 
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
