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1 Answer
MetaPhysik
Jan 3, 2018

Meanlife #= -1/k#

Explanation:

Technically this is a math problem because it involves integration only.

Let

# M= kI#

# I = int_0^oo te^(kt)dt#

Perform integration by parts on I:
#u=t, dv= e^(kt)dt rArr v = 1/ke^(kt)#
#I= uv-vdu =1/k (te^(kt))_0^oo-1/kint_0^oo e^(kt)dt#
#I= uv-vdu =1/k (te^(kt))_0^oo-1/k^2(e^(kt))_0^oo#

#I= 1/k [(te^(-|k|t))_0^oo-1/k(e^(-|k|t))_0^oo]#

Because# lim_(t->oo) te^(-|k|t) ->0 #
For a proof, see https://socratic.org/questions/can-you-find-the-limit-lim-xrarroo-xe-x-2?source=search

#and k= -|k| rArr e^(-k|t|) -> 0#, when # t->oo #

#I = 1/k( (0-0)-1/k(0-1))= 1/k^2 #

#M=-kI = -k(1/k^2) = -1/k = 1/|k| = 1/0.000121#

M= 8264 yrs

To verify this answer, compute C-14 half-life:

Half-life = #ln(2)M = 5729 yrs#

and compare it to the standard C-14 half-life = # 5730 +- 40 yrs #

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