Hyper-intriguing Ornament V - A trapped hyper-bug

Geometry Level pending

Consider a hyper-ball x 2 + y 2 + z 2 + w 2 ≤ 25 x^2 + y^2 + z^2 + w^2 \leq 25 and a concentric hyper-cube [ − 4 , 4 ] 4 \left[-4,4 \right]^4 in R 4 \mathbb{R}^4 . A hyper-bug is stuck in the ornament. What is the probability of finding the hyper-bug only in the hyper-cube part of the ornament? Round your answer, in percentage, to the nearest integer.


The answer is 31.

Huan Bui by Huan Bui , 23, USA

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1 solution

Otto Bretscher
Dec 31, 2018

At this point, it is all just a matter of "crunching the numbers". Let's summarize what we found in problem I, at the beginning of the saga. If B B is the volume of the ball and C C is the volume of the cube, then C = 8 4 = 4096 , B ≈ 3084 , B C = 8 V c a p s ≈ 379 , B ∩ C ≈ 2705 , C B = 1391 , B ∪ C = 4475 C=8^4=4096,\ B\approx 3084,\ B\text\ C=8V_{caps}\approx 379,\ B\cap C\approx2705,\ C\text\ B=1391,\ B \cup C= 4475 The probability we seek is C B B ∪ C ≈ 31 % \frac{C\text\ B}{B \cup C}\approx \boxed{31}\text\%

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Thanks, Otto, for the solution!

Huan Bui - 2 years, 5 months ago

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