HMMT 二月 2009 · GEN2 赛 · 第 4 题

HMMT February 2009 — GEN2 Round — Problem 4

个人笔记

[ 3 ] A torus (donut) having inner radius 2 and outer radius 4 sits on a flat table. What is the radiusof the largest spherical ball that can be placed on top of the center torus so that the ball still touchesthe horizontal plane? (If the x – y plane is the table, the torus is formed by revolving the circle in thex – z plane centered at (3 , 0 , 1) with radius 1 about the z axis. The spherical ball has its center on thez -axis and rests on either the table or the donut.)
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解析

[ 3 ] A torus (donut) having inner radius 2 and outer radius 4 sits on a flat table. What is the radiusof the largest spherical ball that can be placed on top of the center torus so that the ball still touchesthe horizontal plane? (If the x − y plane is the table, the torus is formed by revolving the circle in thex − z plane centered at (3 , 0 , 1) with radius 1 about the z axis. The spherical ball has its center on thez -axis and rests on either the table or the donut.)
Answer: 9 / 4
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Solution: Let r be the radius of the sphere. One can see that it satisfies ( r + 1) = ( r − 1) + 3 by the Pythagorean Theorem, so r = 9 / 4.
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