 All about flooble | fun stuff | Get a free chatterbox | Free JavaScript | Avatars  perplexus dot info  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) Comments: ( Back to comment list | You must be logged in to post comments.) Solution | Comment 3 of 7 | `Divide the square in two with a diagonaland double the the area of the resultingspherical triangle.`
`Area of square shadow = two times area of                        triangle shadow`
`Area of spherical triangle `
`    = R^2*( A' + B' + C' - pi )`
`Where A', B', and C' are the angles between arcs of great circles. Let B' be the imageof the angle of the square that was notdivided by the diagonal. Then,`
`    A' = C' = B'/2`
`Therefore, Area of spherical triangle `
`    = R^2*( 2*B' - pi )`
`The angle B' is the angle between the facesof the square pyramid whose apex is thecenter of the sphere and whose base is thesquare. It is given by the following:`
`    arccos[ -s^2/(s^2 + 4h^2) ]`
`Where h = R-L which is the height of thepyramid. Therefore, the area of the squareshadow is`
`    2*R^2*[2*arccos(-s^2/[s^2 + 4*(R-L)^2]) - pi]`
` `

 Posted by Bractals on 2009-08-19 18:14:21 Please log in:
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