Difference between revisions of "2021 Fall AMC 10A Problems/Problem 19"
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==See Also== | ==See Also== | ||
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Revision as of 20:21, 25 November 2021
. A disk of radius 
rolls all the way around in the inside of a square of side length 
and sweeps out a region of area 
. A second disk of radius rolls all the way around the outside of the same square and sweeps out a region of area 
. The value of 
can be written as 
, where 
and 
are positive integers and 
and are relatively prime. What is 
Solution 1
The side length of the inner square traced out by the disk with radius is 
. However, there is a little triangle piece at each corner where the disk never sweeps out. The combined area of these 
pieces is 
. As a result, 
.
Now, we consider the second disk. The part it sweeps is comprised of quarter circles with radius 
and rectangles with a side lengths of and . When we add it all together, 
. 
so 
. Finally, 
.
Solution 2
The area of the region covered by the first disk is
The area of the region covered by the second disk is
These two equations jointly imply 
.
Therefore, the answer is 
.
~Steven Chen (www.professorchenedu.com)
See Also
| 2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 18 |
Followed by Problem 20 | |
| 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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
