Last spring equinox, brothers Doug
and Ryan Euclid were standing
at the equator looking out to sea.
Doug was standing on the shore,
while Ryan was standing above him
on top of a cliff. Their eyes were 2m
and 23m above sea level, respectively. When Doug saw the sun set, he called out; 26 seconds later
Ryan observed the sun setting.
Doug and Ryan were observant
and knew the time it takes the
Earth to rotate once. The brothers
estimated the diameter of the
Earth.
What was their estimate to the nearest 100km?
In 26 seconds, the earth will rotate by an angle of a = 26/(24*60*60)*2pi radians.
Assuming it's close to the equinox so we are looking due west, we aren't doing anything with refraction in the atmosphere, and the sun is a point on the horizon.
We can form a right triangle with the observer A, the center of the earth B, and the point on the horizon C.
Call earth radius x.
For Doug cos(angleABC)=r/(r+.002)
For Ryan cos(angle A'BC')=r/(r+.023)
The difference of these angles should be the same as angle a.
Thus the equation to solve is
arccos(x/(x+.023))-arccos(x/(x+.002))=26/(24*60*60)*2pi
Unfortunately the solution x=6397 which is well within 100km of Wikipedia's listing of the equatorial radius of 6378km.
My guess is the creator of the problem worked backwards to choose the numbers.
Graph used to solve:
https://www.desmos.com/calculator/5hcjsjfqw1
|
|
Posted by Jer
on 2024-12-14 10:50:38 |
Solution?
