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172 geometry/corner.p




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This article is from the Puzzles FAQ, by Chris Cole chris@questrel.questrel.com and Matthew Daly mwdaly@pobox.com with numerous contributions by others.

172 geometry/corner.p


A hallway of width A turns through 90 degrees into a hallway of width
B. A ladder is to be passed around the corner. If the movement is
within the horizontal plane, what is the maximum length of the ladder?

geometry/corner.s

------\---------
B   /  \.......|..B/sin(theta)
   theta\      |
---|-----X     |
	 |\    |
	 | \...|..A/cos(theta)
	 |  \  |
	 |   \ |
	 | A  \|

Theta is the angle off horizontal.

Minimize length = A/cos(theta) + B/sin(theta)

    d(length)/d(theta)
	   = A*sin(theta)/cos(theta)^2 - B*cos(theta)/sin(theta)^2 (?)
	   = 0
    A*sin(theta)/cos(theta)^2 = B*cos(theta)/sin(theta)^2
 
    B/A = sin(theta)^3/cos(theta()^3 = tan(theta)^3
 
    theta = inverse_tan(cube_root(B/A))

If you use the trigonometric formulas cos^2 x = 1/(1 + tan^2 x)
and sin x = tan x cos x, and plug through the algebra, I believe
that the formula for the length reduces to

(A^(2/3) + B^(2/3))^(3/2)

At any rate this is symmetric in A and B as one would expect, and
has the right values at A=B and as either A-->0 or B-->0.

 

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