## Calculus 10th Edition

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%