Given:
Decay rate: 1.5% per year = -0.015
To find:
The half life.
Solution:
The continuous exponential decay function is
[tex]A(t)=Ae^{-kt}[/tex]
where, a is initial value, -k is decay rate and t is time period.
For half life, [tex]A(t)=\dfrac{A}{2}[/tex],
[tex]\dfrac{A}{2}=Ae^{-0.015t}[/tex]
Dividing both sides by A.
[tex]\dfrac{1}{2}=e^{-0.015t}[/tex]
Taking natural log on both sides.
[tex]\ln \dfrac{1}{2}=\ln e^{-0.015t}[/tex]
[tex]-0.69314718=-0.015t[/tex]
Divide both sides by -0.015.
[tex]\dfrac{-0.69314718}{-0.015}=t[/tex]
[tex]46.209812
=t[/tex]
[tex]t\approx 46[/tex]
Therefore, the half life is 46 years.
