It’s common knowledge that the length of a day (time from sunrise to sunset) varies over the year and with one’s location on the earth. Given only the following, can you derive an approximate equation for the length of a day (in hours) of any specific location on the earth, on any given day of the year?
1) The inclination of the earth’s rotational axis is 23.45 degrees
2) The length of a day is exactly 24 hours
3) The length of a year is exactly 365 days
4) Location on earth is given by the latitude
You are allowed to ignore secondary (but real!) effects such as the earth’s non-circular orbit, the sun being a disc, refraction of sunlight by the atmosphere, etc. To allow easier comparison of different solutions, let’s also assume that North Latitude is positive, and that the Winter Solstice in the Northern Hemisphere is “day 0” (Hint: therefore also day 365!) of the year.
What if the magic bunny either disappears or splits into three identical magic bunnies?
Again, let p = probability that you will eventually be left with no magic bunnies.
Then, p = (1/2)*1 + (1/2)*p^3
p^3 - 2p +1 = 0
Roots are 1, and (-1+/- sqrt(5)/2
What can we make of this?
The negative root is clearly not our desired probability. Somewhat less clearly, neither is p = 1. On average, a magic bunny is expected to turn into 1.5 magic bunnies every night. This is not a process that is guaranteed to eventually result in no bunnies.
No, the answer is (-1+sqrt(5))/ 2 = Phi - 1 = 1/Phi = approx .618034. Phi is the Golden Ratio!
61.8% of the time we are left with no bunnies.
38.2% of the time we wind up, I think, with an infinite number of bunnies.
Edited on August 6, 2024, 7:49 am
The Golden Ratio appears
