A muon is an elementary radioactive particle created when cosmic rays crash into the upper atmosphere of the Earth. The half life of a muon is `1.5*10^-6` s.
c represents the speed of light and is equal to 299792458 m/s. The speed in m/s of a muon moving at a speed equal to 0.995c is 0.995*299792458 = 298293495.71 m/s
The distance traveled by a muon traveling at 0.995c in one microsecond can be determined using the formula D = T*S where D is the distance traveled in time T by an object moving at speed D. Substituting the values given the muon travels a distance equal to `1*10^-6*298293495.71` = 298.29349571 m.
The height of Mount Washington is 6300 ft. One ft is equal to 0.3048 m. 6300 ft is equal to 1920.24 m. The time taken by a muon traveling at 0.995c to travel 1920.24 m is T = `1920.24/298293495.71 = 6.437*10^-6` s or approximately 6.4 `mu*s`
If there are initially `N_0` particles, the number of particles after time t is given by `N = N_0/2^(t/t_g)` where `t_g` is the half life. Initially 568 muons are detected, the number left after 6.4 `mu*s` is equal to `568/2^((6.4*10^-6)/(1.5*10^-6))` = `568/2^(6.4/1.5) ~~ 29`