SOLUTION: 8. Radioactive materials like uranium follow the law of uninhibited decay, which is an Exponential Model. This decay causes radiation. All radioactive substances have a specific ha

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: 8. Radioactive materials like uranium follow the law of uninhibited decay, which is an Exponential Model. This decay causes radiation. All radioactive substances have a specific ha      Log On


   

Question 825694: 8. Radioactive materials like uranium follow the law of uninhibited decay, which is an Exponential Model. This decay causes radiation. All radioactive substances have a specific half-life, which is the time required for half of the radioactive substance to decay. Uninhibited Radioactive Decay is given by the formula:

The half-life of thorium-229 is 7,340 years. How long will it take for a sample of this substance to decay to 20% of its original amount?
Answer by josgarithmetic(39838) About Me  (Show Source):
You can put this solution on YOUR website!
The exponential decay model is based on A=I%2Ae%5E%28kt%29; some people might prefer a -k, but however the choice, consistency is needed.

A = amount at time t
I = initial amount, for t=0
k = a constant
t = time in years

First, find k.
A%2FI=e%5E%28kt%29
ln%28A%2FI%29=ln%28e%5E%28kt%29%29
ln%28A%2FI%29=kt%2A1
k=%281%2Ft%29ln%28A%2FI%29
'
The given halflife is for t=7340 years, and for which A=%281%2F2%29I;
k=%281%2F7340%29ln%281%2F2%29
k=-9.44%2A10%5E%28-5%29
'
Not sure if this will render properly but the model is:
highlight%28A=I%2Ae%5E%28-0.0000944%2At%29%29

Your specific question will use A=0.20, and I=1. You would then solve the model equation for t.
Recall from, "First, find k", a slight change in step gives: highlight%28t=%281%2Fk%29ln%28A%2FI%29%29; just substitute the values known.