Difference between revisions of "1987 AIME Problems/Problem 2"
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== Problem == | == Problem == | ||
| − | What is the largest possible [[distance]] between two [[point]]s, one on the [[sphere]] of [[radius]] 19 with [[center]] <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 == | ||
| − | 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 = 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>. |
== 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}} | ||
Latest revision as of 17:46, 14 February 2014
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
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 
.
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. 
