Question

Please help me this question?

Answer 1

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`