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}</cmath> This implies our answer is <math>225+73=\boxed{298}</math>. ~mn28407
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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}\imples225+73=\boxed{298}</cmath>~mn28407
  
 
==Video Solution==
 
==Video Solution==

Revision as of 14:26, 8 June 2020

Problem

Two congruent right circular cones each with base radius $3$ and height $8$ have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance $3$ from the base of each cone. A sphere with radius $r$ lies withing both cones. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$n and $n$ are relatively prime positive integers. Find $m+n$.

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, $O$, and the bases form the positive $x$ and $y$ axes. Then label the vertices of the region enclosed by the two triangles as $O,A,B,C$ in a clockwise manner. We want to find the radius of the inscribed circle of $OABC$. By symmetry, the center of this circle must be $(3,3)$. $\overline{OA}$ can be represented as $8x-3y=0$ Using the point-line distance formula,

\[r^2=\frac{15^2}{(\sqrt{8^2+3^2})^2}=\frac{225}{73}\imples225+73=\boxed{298}\] (Error compiling LaTeX. ! Undefined control sequence.)

~mn28407

Video Solution

https://youtu.be/bz5N-jI2e0U?t=44

See Also

2020 AIME II (ProblemsAnswer KeyResources)
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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