Distance and Spheres | AIME I, 1987 | Question 2
Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1987 based on Distance and Spheres.
Distance and Sphere - AIME I, 1987
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)?
- is 107
- is 137
- is 840
- cannot be determined from the given information
Key Concepts
Angles
Algebra
Spheres
Check the Answer
Answer: is 137.
AIME I, 1987, Question 2
Geometry Vol I to Vol IV by Hall and Stevens
Try with Hints
The distance between the center of the spheres is \(\sqrt{(12-(-2)^{2}+(8-(-10))^{2}+(-16-5)^{2}}\)
=\(\sqrt{14^{2}+18^{2}+21^{2}}\)=31
The largest possible distance=sum of the two radii+distance between the centers=19+87+31=137.
Other useful links
- https://cheenta.com/cubes-and-rectangles-math-olympiad-hanoi-2018/
- https://www.youtube.com/watch?v=ST58GTF95t4&t=140s