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AIME 1987 · 第 2 题

AIME 1987 — Problem 2

专题
Contest Math
难度
L4
来源
AIME

题目详情

Problem

What is the largest possible distance between two points, one on the sphere of radius 19 with center (−2,−10,5)(-2,-10,5) and the other on the sphere of radius 87 with center (12,8,−16)(12,8,-16)?

解析

Solution 1

The distance between the two centers of the spheres can be determined via the distance formula in three dimensions: (12−(−2))2+(8−(−10))2+(−16−5)2=142+182+212=31\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \sqrt{14^2 + 18^2 + 21^2} = 31. The largest possible distance would be the sum of the two radii and the distance between the two centers, making it 19+87+31=13719 + 87 + 31 = \boxed{137}.

Solution 2

Since you have a lot of time on the AIME, you could spend 3 hours drawing a paper 3-dimensional graph of the two circles. Then, you would guess and check two random points on the circles until you get the two farthest points, and you find the distance will be 137\boxed{137}. (However, this could be very inaccurate and may take up a lot of time, but very much recommended :)