Will you help me with this geometry question, please?

Question

Will you help me with this geometry question, please?
Thank you a bunch.
And show/explain how to get the result?

An astronaut is tethered to the corner of a solid cube with edges of length 50 (on the outside) by a rope of length 10. Assuming there is no gravity, what is the volume of the space in which the astronaut is free to roam?

Thanks a bunch. <3

Answer 1

::
The volume of the sphere of radius 10 is

4/3pi(10)^3

The volume occupied by the corner of the cube where he is tethered (where he could not roam) is 1/8 of the sphere. Therefore the space where he could roam is 7/8 of the sphere which is

= (7/8)*4/3pi(10)^3

= 3500/3pi cu units

Answer 2

the space that the astronaut can roam is that of a sphere whose volume is:

V = (4 Pi/3)r^3
V = 1.3333 pi (10)^3
V = 1333.33 pi
Now the most the cube will take away form that space is a shape of a pyramid with a spherical base and 3 sides each 10 long.
Volume of a pyramid + volume of spherical cap = volume not available to astronaut. So the astronaut has 7/8 of 1333.33pi because the cubes corner takes up 1/8 of his space so the final answer is:

7/8 * 1333.333 pi cubic units.
=` 1166.6667 pi cubic units.

Answer 3

I think..okay.

The space the astronaut can roam would be the volume enclosed in a sphere with the radius as 10 & the center of the sphere would be the corner of the cube that the rope is attached to..

The way I think it would look(I warn you, it's a bad drawing haha):
http://img516.imageshack.us/img516/9607/bahahazl6.png

So, the volume of a sphere
V=4/3piR^3
=(4/3)pi1000
=4 188.7902 cubic units

Check with other people, tho. This is what I think, but Idk if it's correct.