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 The Shadow in a Sphere (Posted on 2009-08-19)
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: Solution | Comment 5 of 7 |
(In reply to Solution by Bractals)

See correction in later post. Bractals' formula is correct.

Using Bractal's formula:

DECLARE FUNCTION acos! (x!)
DECLARE FUNCTION asin# (x#)
DEFDBL A-Z

DIM SHARED pi
pi = ATN(1) * 4

CLS
DO
INPUT "r,s,l:", r, s, l
a = 2 * r ^ 2 * (2 * acos(-s * s / (s ^ 2 + 4 * (r - l) ^ 2)) - pi)
PRINT a
LOOP

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

DEFDBL A-Z
FUNCTION asin (x)
c = SQR(1 - x * x)
asin = ATN(x / c)
END FUNCTION

gives

r,s,l:1,.2,.4
-12.45824939409365
r,s,l:2,.3,.5
-50.10706450142342

Definitely shouldn't be negative, and disagrees calculations using my formula.

Edited on August 20, 2009, 12:30 am
 Posted by Charlie on 2009-08-19 19:29:41

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