# Determine the amount of radioactive element left after five years if you know it follows the law `N = N_(o)e^(-kT)` where T is measure in years.  From previous measurements we know that the original amount of radioactive material was 100 grams and after two years there remained 80 grams.

The first step to answer this problem is to determine the decay constant "k" :

`N = N_oe^(-kT)` divide both sides by `N_o` and applying the symmetric property of equality gives

`e^(-kT) = N/N_o` we can then take the natural log of both sides

`ln(e^(-kT)) = ln(N/N_o)` which produces `-kT = ln(N/N_o)` therefore

`k = -(ln(N/N_o))/T` we can now substitute the information for two years and solve for k

`k = -(ln(80/100)/2) = 0.113`

This gives the general equation for N of

`N = 100 e^(-0.113T)` .  Now substituting 5 years for T we get

`N = 100e^(-0.565) = 56.8 grams`

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