1987 AIME Problems/Problem 2

Problem

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

Solution

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

See also

1987 AIME (ProblemsAnswer KeyResources)
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. AMC logo.png

Invalid username
Login to AoPS