



Sometimes
how fast something gets small



also
depends on how much there is. 


One
of the times this happens with something called radio active decay. 


When
radio active stuff decays, 


it
crumbles into stuff too small to see 


and
energy. 


The
time it takes for half the radio active stuff to crumble away
(decay) 


is
called it's HALF LIFE. 





When
we figure out stuff like this 


the
equation is the same as the ones we saw on the last page. 





P = P_{o}e^{kt} 





Example: 


An
unknown radio active substance loses 10% of it's mass 


to
decay in one year. 


How
long will it take to lose half of it's mass to radioactive decay? 





First
we need to find k ... 






P = .9 



P_{o
= }1 



t = 1 



k = the
unknown 





P = P_{o}e^{kt} 


.9 = 1 × e^{k
× 1} 


.9 = e^{k} 





Now
use the ln trick ... 




ln .9 = ln e^{k} 




ln .9 = k (ln
.9 = .105) 




 .105 = k 





Now
that we know what k is, we can solve the problem. 


The
problem is: 


"How
long will it take for it to lose half of it's mass 


to
radioactive decay?" 


So: 



P = .5 



P_{o
= }1 



k =
.105 



t = the
unknown 





Now
we can do the problem ... 


P = P_{o}e^{kt} 


.5 = 1 e^{.105t} 





Now
we use the log trick 




ln .5 = ln e^{.105t} 




.693 = .105t 





Now
we can divide both sides by .105 to get t by itself ... 











So
in 6.6 years, half of our radioactive stuff will have decayed. 


That
means the half life of this stuff is 6.6 years. 


That
is, no matter when you look at the stuff, 


6.6
years later there will be half as much there. 





Closing
Notes: 


Any
time P is bigger than Po we have growth and k turns out positive 


Any
time P is smaller than Po we have decay and k turns out negative. 





copyright 2005 Bruce Kirkpatrick 
