How Much Empty Space

Geometry Level 5

A red sphere of radius 8 cm fits snugly in a cubical box. 8 green spheres fit snugly in the corners of the box. To the nearest integer, find the volume of empty space in this box.

P.S. Imagine the other green spheres are included in the diagram.


The answer is 1621.

Guiseppi Butel by Guiseppi Butel , 90, Canada

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4 solutions

Guiseppi Butel
Jun 12, 2014

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You have really cool handwriting. :D

Finn Hulse - 7 years ago

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Thanks!

What did you think of the problem?

Guiseppi Butel - 7 years ago

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It was excellent but I somehow got a wrong answer... Computation error. :P

Finn Hulse - 7 years ago

I thought the 8cm is the side length of the cube.. Wested my first two chances. Nice question BTW.

Sanjeet Raria - 6 years, 8 months ago

1) How is RB = 2 x^2 2) How does OB^2 = x^2+RB^2 if it is not a right angle? 3) Shouldn't EB = x + x sqrt(2)?

Timothy Tiong - 6 years, 8 months ago

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Sorry got it now. I was thinking in 2D. Will retry again.

Timothy Tiong - 6 years, 8 months ago

radius 'r' of the smaller green sphere is given as r = R ( 2 − 3 ) \LARGE r= R(2-\sqrt {3}) Hence the volume of the void space = 2 R 3 − 4 π 3 R 3 − 8 × 4 π 3 ( R ( 2 − 3 ) ) 3 \LARGE ={2R}^3 -\frac{4 \pi}{3} {R}^3-8 \times \frac{4\pi}{3} \left(R(2-\sqrt{3})\right)^3 Substituting the value R=8cm in the above expression you will get the answer.

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@Guiseppi Butel , I really admire your problems involving geometry, especially this one.. Sad to say that it's already late when I learned about that kissing circles problem.. :3 but still, I'm striving hard so as to gain advanced and more learning like you do, and I'm studying more things in math that I still lack.. :)

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Rudresh Tomar
Oct 25, 2014

Its a sangaku question...check my sets!

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It's a good problem, but not a level 5 problem imo

Nowroze Farhan - 1 year, 2 months ago

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