Difference between revisions of "1987 AIME Problems/Problem 2"
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What is the largest possible [[distance]] between two [[point]]s, one on the [[sphere]] of [[radius]] 19 with [[center]] <math>(-2,-10,5)</math> and the other on the sphere of radius 87 with center <math>(12,8,-16)</math>? | What is the largest possible [[distance]] between two [[point]]s, one on the [[sphere]] of [[radius]] 19 with [[center]] <math>(-2,-10,5)</math> and the other on the sphere of radius 87 with center <math>(12,8,-16)</math>? | ||
| − | == Solution == | + | == Solution 1 == |
The distance between the two centers of the spheres can be determined via the [[distance formula]] in three dimensions: <math>\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \sqrt{14^2 + 18^2 + 21^2} = 31</math>. The largest possible distance would be the sum of the two radii and the distance between the two centers, making it <math>19 + 87 + 31 = \boxed{137}</math>. | The distance between the two centers of the spheres can be determined via the [[distance formula]] in three dimensions: <math>\sqrt{(12 - (-2))^2 + (8 - (-10))^2 + (-16 - 5)^2} = \sqrt{14^2 + 18^2 + 21^2} = 31</math>. The largest possible distance would be the sum of the two radii and the distance between the two centers, making it <math>19 + 87 + 31 = \boxed{137}</math>. | ||
| + | == 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 <math>\boxed{137}</math>. (However, this could be very inaccurate and may take up a lot of time, but very much recommended :) | ||
== See also == | == See also == | ||
{{AIME box|year=1987|num-b=1|num-a=3}} | {{AIME box|year=1987|num-b=1|num-a=3}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Latest revision as of 21:53, 3 February 2025
Contents
Problem
What is the largest possible distance between two points, one on the sphere of radius 19 with center 
and the other on the sphere of radius 87 with center 
?
Solution 1
The distance between the two centers of the spheres can be determined via the distance formula in three dimensions: 
. The largest possible distance would be the sum of the two radii and the distance between the two centers, making it 
.
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 
. (However, this could be very inaccurate and may take up a lot of time, but very much recommended :)
See also
| 1987 AIME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
