Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Distance and Spheres.
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)?
Angles
Algebra
Spheres
But try the problem first...
Answer: is 137.
AIME I, 1987, Question 2
Geometry Vol I to Vol IV by Hall and Stevens
First hint
The distance between the center of the spheres is \(\sqrt{(12-(-2)^{2}+(8-(-10))^{2}+(-16-5)^{2}}\)
Second Hint
=\(\sqrt{14^{2}+18^{2}+21^{2}}\)=31
Final Step
The largest possible distance=sum of the two radii+distance between the centers=19+87+31=137.
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