Answer
(a) The half-life is 12.3 years.
(b) 28.5 years
Work Step by Step
(a) We can find the value of $k$:
$m(t) = m(0)e^{kt}$
$m(1) = m(0)e^{k} = 0.945~m(0)$
$e^{k} = 0.945$
$k = ln(0.945)$
$k = -0.05657$
Then:
$m(t) = m(0)~e^{-0.05657~t}$
We can find the time $t$ when only half remains:
$m(t) = m(0)~e^{-0.05657~t} = 0.5~m(0)$
$e^{-0.05657~t} = 0.5$
$(-0.05657)~t = ln(0.5)$
$t = \frac{ln(0.5)}{-0.05657}$
$t = 12.3~years$
The half-life is 12.3 years.
(b) We can find the time $t$ when only 20% remains:
$m(t) = m(0)~e^{-0.05657~t} = 0.2~m(0)$
$e^{-0.05657~t} = 0.2$
$(-0.05657)~t = ln(0.2)$
$t = \frac{ln(0.2)}{-0.05657}$
$t = 28.5~years$