2021 Fall AMC 10A Problems/Problem 19
Problem
A disk of radius rolls all the way around 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, .
~MathFun1000 (Inspired by Way Tan)
See Also
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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