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Hinged Screen

 
Bert Bates
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A hinged screen with two panels of the same size is placed into the (90 degree) corner of a room. At what angle should the panels be arranged so that the corner and the screen enclose the greatest area?

[ November 16, 2003: Message edited by: Bert Bates ]
 
Jim Yingst
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Hmmmm, a bit more complex than I first thought. I'm getting careless. Anyway, the angle with the wall is 3π/8, or 67.5°
 
Joel McNary
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Hmm.
If the width of one panel is x, and the angle of the hinge is 90 degrees, then the area enclosed will be x^2, for you will have formed a square of side length x.
If you make the angle of this hinges 180 degrees (the two panels form a straight line), then you can make a 45-45-90 right triangle in the corner with a base of 2x and legs x * sqrt(2), creating a triangle with base 2x and height x, and thus having an area of x^2 (same as the square).
However, if you have an angle of 135 degrees, then you have created a shape that can be described as two trianlges that share the same base. That base is of length approx. 1.86 * x, and the two trangles have areas of .35 x ^2 and .85 x ^2, summing for an area of 1.2 x ^2
Since both extremes have been demonstrated to have the same value, (and since I never did do much with calculus), I will take as the maximum possible the midpoint between the two extremes (135 degrees), with an area of 1.2x^2.
 
Joel McNary
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Which does form an angle with the wall of 67.5 degrees.
 
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