Difference between revisions of "2020 AIME II Problems/Problem 7"
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==Solution== | ==Solution== | ||
− | Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, <math>O</math>, and the bases form the positive <math>x</math> and <math>y</math> axes. Then label the vertices of the region enclosed by the two triangles as <math>O,A,B,C</math> in a clockwise manner. We want to find the radius of the inscribed circle of <math>OABC</math>. By symmetry, the center of this circle must be <math>(3,3)</math>. <math>\overline{OA}</math> can be represented as <math>8x-3y=0</math> Using the point-line distance formula, <cmath>r^2=\frac{15^2}{(\sqrt{8^2+3^2})^2}=\frac{225}{73}\ | + | Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, <math>O</math>, and the bases form the positive <math>x</math> and <math>y</math> axes. Then label the vertices of the region enclosed by the two triangles as <math>O,A,B,C</math> in a clockwise manner. We want to find the radius of the inscribed circle of <math>OABC</math>. By symmetry, the center of this circle must be <math>(3,3)</math>. <math>\overline{OA}</math> can be represented as <math>8x-3y=0</math> Using the point-line distance formula, <cmath>r^2=\frac{15^2}{(\sqrt{8^2+3^2})^2}=\frac{225}{73}\implies225+73=\boxed{298}</cmath>~mn28407 |
==Video Solution== | ==Video Solution== |
Revision as of 14:27, 8 June 2020
Contents
Problem
Two congruent right circular cones each with base radius and height have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance from the base of each cone. A sphere with radius lies withing both cones. The maximum possible value of is , where n and are relatively prime positive integers. Find .
Solution
Take the cross-section of the plane of symmetry formed by the two cones. Let the point where the bases intersect be the origin, , and the bases form the positive and axes. Then label the vertices of the region enclosed by the two triangles as in a clockwise manner. We want to find the radius of the inscribed circle of . By symmetry, the center of this circle must be . can be represented as Using the point-line distance formula, ~mn28407
Video Solution
https://youtu.be/bz5N-jI2e0U?t=44
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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