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Quiz of the Month (September 2004)

Hector C. Parr

***

SOLUTION TO LAST MONTH'S QUIZ

   1.  81.32o
   2.  0.4245
   3.  72.43o

Notes
 1. cos x = 6400 / (6400+36000) (etc.)

 2. It is well-known that the area cut off on the surface
    of a sphere between two parallel planes equals the
    area cut off on the circumscribing cylinder.
    So fraction = BC/AC = (r - r cos x)/2r = (1 - cos x)/2

 3. S = point directly under satellite no.1
    N = most northerly point from which both visible.
    Then in the spherical triangle QNS,
      Angle Q = 90o
      Side NS = x = 81.32o
      Side QS = 60o
       Let y = Side NQ = latitude of N
      Then cos x = cos y * cos 60o (by Spherical Trigonometry)
      So cos y = 2 cos x (etc.)

    (Alternatively project ON and OQ onto the tangent plane at S.)

THIS MONTH'S QUIZ

Each step of a spiral staircase has a rise of 20 cm, and the thirteenth step is directly above the first step. The radius of the central wall is 40 cm, and of the outer wall is 104 cm, as shown.
1. Find the area of one tread.

2. A soldier has been ordered always to carry his spear horizontally. What is the length of the longest spear he can take up the stairs?

3. If he is allowed to tilt his spear, then what is the length of the longest spear he can take up the stairs?

© Hector C. Parr (2004)

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