Torus: radius of generating circle

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March 7, 2013 - 10:13pm
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HELLectrical
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Torus: radius of generating circle

A torus is generated by revolving a circle about a line. If the volume generated is numerically equal to en times its surface area, find the radius of the rotating circle.

pls. help me to solve this. Thanks!.

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March 7, 2013 - 10:24pm
Romel Verterra's picture
Romel Verterra
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Joined: Oct 12 2008 - 3:39pm
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Re: Torus

Let
R = distance from the axis of revolution (line) to the center of the circle
r = radius of the circle
 

Area of circle
$ A_c = \pi r^2 $
 

Circumference of circle
$ c = 2\pi r $
 

Volume of torus
$ V = A_c \times 2\pi R $

$ V = \pi r^2 \times 2\pi R $

$ V = 2\pi^2 Rr^2 $
 

Surface area of torus
$ A = c \times 2\pi R $

$ A = 2\pi r \times 2\pi R $

$ A = 4\pi^2 Rr $
 

$ V = 10A $

$ 2\pi^2 Rr^2 = 10(4\pi^2 Rr) $

$ 2r = 40 $

$ r = 20 ~ \text{units} $

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