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Bully For You (Posted on 2004-12-02) Difficulty: 4 of 5

Take a regular torus (doughnut) shaped object and cut a vertical slice through it at line A-A1(Fig1).

Now look at the cross section formed(Fig2). Is it possible to calculate the volume of the original Torus? Prove your results.

No Solution Yet Submitted by Juggler    
Rating: 3.6667 (3 votes)

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Full solution | Comment 1 of 5

The answer is yes.  (The approximate volume is 18606 cubic millimeters)

Here's what I did:  (This is really hard to explain)

Consider the plane perpendicular to A-A1 whose cross section is a circle of diameter 12.

Use to pythagorean theorem to find the distance from the outer edge of this circle to the point where plane A-A1 cuts it.  This distance is 6 + sqrt(35).  [35 = 6^2 - 1^2]

Consider a top view of this torus.

There is a right triangle (actually two) formed by the two plane sections and the points where they intersect the major diameter of the torus.

The legs of this triangle are 25 and 6+sqrt(35). Use the inverse-tangent to find the larger acute angle.  It is about 64.5155 degrees.

Connect the ends of the hypotenuse to the center of the torus to form a isoceles triangle with its vertex at the center.  This vertex angle can be found by 180 - 2*64.5155 = 50.967

There is also a right triangle with leg = 25 and angle = 50.967.  The hypotenuse of this triangle is the outer radius (R) of the torus.  Use sin(50.967) = 25/R to find R = 32.183

r = R - 12 = 20.183

V = .25*pi^2(R+r)(R-r)^2  = 18606.01786

[corrected volume formula] [twice]

Exact R = 25/(sin(180-2arctan(25/(6+sqrt(35)))))

Edited on December 2, 2004, 6:11 pm

Edited on December 2, 2004, 8:09 pm
  Posted by Jer on 2004-12-02 18:05:24

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