The given equation is
m' = -0.01m
where m = the mass at time t, days.
Assume that m(0) = m₀, the initial mass.
The solution for the differential equation is as follows:
[tex] \frac{dm}{dt} = -0.01m \\\\ \frac{dm}{m} = -0.01dt \\\\ \int _{m_{0}}^{m} \frac{dm}{m} = -0.01 \int_{0}^{t} dt \\\\ ln \frac{m}{m_{0}} =-0.01t \\\\ m(t)=m_{0} e^{-0.01t} [/tex]
At half-life, the time is given by
[tex]e^{-0.01t} = \frac{m_{0}/2}{m_{0}} = \frac{1}{2} \\\\ -0.01t = ln(0.5) \\\\ t = - \frac{ln(0.5)x}{-0.01} = 69.315[/tex]
Answer: The half-life is 69.3 days.
