Solution 3.1

    The probability of a ray hitting the one meter square picnic blanket is the ratio of the shadow that the picnic blanket would make on a sphere centered at the sun to the surface of the entire sphere.

  1. The surface of a sphere with radius 1.5e11 meters is approximately 1.41e34, so the probability of a particular ray hitting the picnic blanket is approximately 1/1.41e34=7.07e-35 when the sun is directly overhead.
  2. When the blanket is tilted at n degrees is the original probability multiplied by cos(n) because this corresponds to the projected shadow onto the sphere surface.
  3. The assumption made was that a 1 metere square from the surface of the sphere of radius 1.5e11 meters is essentially flat. Therefore while the picnic blanket does not exactly match the curvature of the sphere, it provides a sufficiently close approximation.
Christopher Jenkins
Last modified: Sun Jan 12 12:13:01 EST 2003