Problem:
What is the largest possible distance between two points, one on the sphere of radius with center and the other on the sphere of radius 87 with center
Solution:
Let and be the centers of the two spheres, and let and be two points where the extensions of the segment pierce the two spheres, respectively, so that is between and , and is between and . Then the desired maximum distance is
Where and are the given radii and is found by the Distance Formula. In our case, .
To see that (1) indeed yields the maximum distance, note that by the Triangle Inequality, for any points and on the spheres with centers and , respectively,
Note. The above solution does not depend on the position of the spheres relative to one another, as can be seen in the two configurations below, showing cross sections of the spheres by planes containing their centers.
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The problems on this page are the property of the MAA's American Mathematics Competitions