Answer
(a) $m(t) = 50~e^{-0.024755~t}$
(b) After 40 days, the remaining mass is $~~18.6~mg$
(c) It takes $~~130~days~~$ for the sample to decay to $2~mg$
(d) We can see the sketch of the mass function below.
Work Step by Step
(a) $m(t) = m(0)e^{kt}$
$m(28) = (50)e^{28k} = 25$
$e^{28k} = 0.5$
$28k = ln(0.5)$
$k = \frac{ln(0.5)}{28}$
$k = -0.024755$
Then:
$m(t) = 50~e^{-0.024755~t}$
(b) We can find the mass after 40 days:
$m(t) = 50~e^{-0.024755~t}$
$m(40) = 50~e^{(-0.024755)~(40)}$
$m(40) = 18.6~mg$
After 40 days, the remaining mass is $~~18.6~mg$
(c) We can find the time it takes the sample to decay to $2~mg$:
$m(t) = 50~e^{-0.024755~t} = 2$
$50~e^{-0.024755~t} = 2$
$e^{-0.024755~t} = 0.04$
$-0.024755~t = ln(0.04)$
$t = \frac{ln(0.04)}{-0.024755}$
$t = 130~days$
It takes $~~130~days~~$ for the sample to decay to $2~mg$
(d) We can see the sketch of the mass function below.
