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The Shadow in a Sphere (Posted on 2009-08-19) Difficulty: 3 of 5
Consider a hollow sphere of radius R, in which a light source is placed at its centre. A square plate of side length S is held in place within the sphere by a pole of length L units. The square plate's position is then such that the displacement between the centre of the square and the light source is R-L units.

The square plate is also oriented in a way such that an imaginary line drawn perpendicular to the surface of the plate and passing through the plate's centre will pass through the light source.

Determine the surface area of the shadow formed on the spherical shell, due to the square plate.

No Solution Yet Submitted by Chris, PhD    
Rating: 4.0000 (4 votes)

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re(2): Solution | Comment 6 of 7 |
(In reply to re: Solution by Charlie)

My apologies to Bractals, and congratulations on having the simpler formula.

The mismatch in answers was caused by my faulty function definition for acos(x).

I had coded it as

FUNCTION acos (x)
 s = SQR(1 - x * x)
 acos = ATN(s / x)
END FUNCTION

However, this gives a negative angle for negative x, which is wrong.

The corrected acos(x) function combined with Bractals' formula gives the correct answers, in agreement with those I found.

FUNCTION acos (x)
 acos = pi / 2 - asin(x)
END FUNCTION

The faulty formula worked correctly for the acute angle I was going after, but incorrectly for the obtuse angle needed in Bractals formula.

Edited on August 20, 2009, 12:55 am
  Posted by Charlie on 2009-08-20 00:33:46

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