# Application of 1st order differential equations 2

• May 19th 2009, 01:02 AM
sanikui
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)
• May 19th 2009, 03:52 AM
Jester
Quote:

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 $\displaystyle \frac{dP}{dt} = - k P,\;P(0)=A$ which you give the solution $\displaystyle P = Ae^{-kt}$. The half life is 1400 yrs so $\displaystyle P(1400) = Ae^{-1400k} = \frac{A}{2}$ from which you can solve for k (the A's cancel). Then find t* such that $\displaystyle P(t^*) = Ae^{-kt*} = .99A$ (again the A's cancel).