Answer
97.856%
Work Step by Step
Interpreting the rate of decomposition as:
$\frac{dy}{dt}= ky$, integrate to find y
$\int \frac{1}{y} dy= \int kdt$
$\ln|y| = kt+C$
$y= Ce^{kt} $
Using the half-life time to be 1599 years
$0.5= e^{1599k} $
$ -\ln{2} = 1599k$
$k= \frac{-\ln {2}}{1599}$
Plug back into the Decomposition equation
$y=Ce^{\frac{-\ln {2}}{1599}t} $
Find the percentage of a present amount after 50 years
$\frac{y}{C} = e^{{\frac{-\ln {2}}{1599}*50}}$
$ \frac{y}{C}= .97855$
percent remaining = 97.856%
