Algebra 2 Exponential Decay
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Where'd It Go?
Exponential Decay

 

 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 = Poekt

 
 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
  Po = 1
  t = 1
  k = the unknown
 

P = Poekt

.9 = 1 ek 1

.9 = ek

 
 Now use the ln trick ...
 

ln .9 = ln ek

 

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
  Po = 1
  k = -.105
  t = the unknown
 
 Now we can do the problem ...

P = Poekt

.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

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