Difference between revisions of "2023 AMC 10B Problems/Problem 24"
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==See also== | ==See also== | ||
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Revision as of 21:06, 15 November 2023
Problem
What is the perimeter of the boundary of the region consisting of all points which can be expressed as with , and ?
Solution
Notice that this we are given a parametric form of the region, and is used in both and . We first fix and to , and graph from :
Now, when we vary from to , this line is translated to the right units:
We know that any points in the region between the line (or rather segment) and its translation satisfy and , so we shade in the region:
We can also shift this quadrilateral one unit up, because of . Thus, this is our figure:
The length of the boundary is simply ( can be obtained by Pythagorean theorem, since we have side lengths and .). This equals
~Technodoggo
Video Solution 1 by OmegaLearn
See also
2023 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 23 |
Followed by Problem 25 | |
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.