3-D Geometry

Forum for the GRE subject test in mathematics.
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3-D Geometry

Postby mobius70 » Mon Mar 10, 2008 7:41 am


Can anyone help me with this.


The surface y^2 + z^2 =1 in 3-space is an example of an (infinite) horizontal right circular cylinder of radius 1; the axis of this particular cylinder is the x-axis. Now suppose we have n horizontal right circular cylinders, all of radius 1, whose axes are all (horizontal) lines through the origin which make equal angles to each other there. (For instance , if n=4, the axes could be the x- and y-axes and the lines y=x and y=-x.)

a. Find the volume that lies within all n cylinders.

b. What happens to your answer from a) as n->infinity? Can you explain this?

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Postby lime » Wed Mar 12, 2008 7:29 pm

First of all, I tried to sketch out how it would look like. For the two cylinders intersection case I was eventually able to produce something like the picture below.
My "tube" looks like a kind of flattened out, but trust me it is round. :wink: I hope the idea is clear.
Apparently, with increasing the number of cylinders this figure will have more and more facets and consequently supposed to turn in the sphere.
The area will consist of eight parts of so called "cylindrical hoof". I found it here with a formula for its volume.
It is obvious, that in case with n cylinders we will have n*4 cylindrical hoofs of the same size. Therefore, according to some relations that should be clear from the picture we can find the volume of the area.
Evaluating the limit with n->infinite, we use the fact that sin(PI/n)~PI/n:
Which is indeed the area of the sphere!!

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Postby ieoi » Sat Nov 01, 2008 4:10 am

lime, good work~ I am impressed.

but, you omitted n in the late equation of first problem.

V = (8/3)R^(3)(n)tan(Pi/2n)

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