# Thread: Application of 1st order differential equations 2

1. ## Application of 1st order differential equations 2

The decaying rate of radium is directly proportional to its access mass.
a) Write a differential equation to relate mass of radium, x and time t.
b) If it takes 1400 years for a sample to lose half of its initial mass, how long does it take to lose 1% of the initial mass?

For question a) x=A(e^-kt) A=e^c

For question b) My problem was the initial mass...actually how to do this type of question??(IF mass was not given)

2. Originally Posted by sanikui
The decaying rate of radium is directly proportional to its access mass.
a) Write a differential equation to relate mass of radium, x and time t.
b) If it takes 1400 years for a sample to lose half of its initial mass, how long does it take to lose 1% of the initial mass?

For question a) x=A(e^-kt) A=e^c

For question b) My problem was the initial mass...actually how to do this type of question??(IF mass was not given)
Actually the differential equation is $\frac{dP}{dt} = - k P,\;P(0)=A$ which you give the solution $P = Ae^{-kt}$. The half life is 1400 yrs so $P(1400) = Ae^{-1400k} = \frac{A}{2}$ from which you can solve for k (the A's cancel). Then find t* such that $P(t^*) = Ae^{-kt*} = .99A$ (again the A's cancel).