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        riddles >> medium >> Two Intersecting Spheres
        
(Message started by: ThudanBlunder on Oct 21st, 2008, 3:00pm)
Title: Two Intersecting Spheres
Post by ThudanBlunder on Oct 21st, 2008, 3:00pm
S and T, with radii r and R respectively, are two intersecting spheres such that S passes through the centre of T. If R http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/leqslant.gif 2r find the area of the surface of S that lies inside T.
Title: Re: Two Intersecting Spheres
Post by Grimbal on Oct 21st, 2008, 3:48pm
Surprising result, looks wrong at first, and yet it works.
Title: Re: Two Intersecting Spheres
Post by towr on Oct 22nd, 2008, 1:45am
Guessing, [hide]pi R2[/hide]?
Works at 3 points, certainly.
[hide]R=0; quite obvious.
R=sqrt(2) r: 2 pi r2, half the sphere S.
R=2 r:  4 pi r2, whole sphere S.[/hide]
Title: Re: Two Intersecting Spheres
Post by SMQ on Oct 22nd, 2008, 6:21am
Good guess, towr!

[hide]Let point O be the center of sphere S and point P be the center of sphere T.  Take any plane passing through both O and P, and let point Q be one of the two points where S and T intersect in the plane.  Triangle OPQ has sides of r, r and R, and so angle POQ has value http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/alpha.gif= 2sin-1(R/2r).

The surface area of S within T is then http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supalpha.gif 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr sin(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) r dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
 = 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr2 http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supalpha.gif sin(http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif) dhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/theta.gif
 = 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr2(1 - cos(2sin-1(R/2r)))
 = 2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr2(1 - (1 - 2sin2(sin-1(R/2r))))
 = 4http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifr2(R/2r)2
 = http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifR2
[/hide]
--SMQ


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