A certain radioactive isotope has leaked into a stream

Question

A certain radioactive isotope has leaked into a small stream. Two hundred days after the leak, 8% of the original amount of the substance remained. Determine the half-life of this radioactive isotope.

How many days is the half life of the substance?

Thanks!

Answer 1

I just use the compound continuously formula for growth and decay, which is A=Pe^rt

 

So 8% left after 200 days is the same as saying you started with 100 and were left with 8 after 200 days.....

 

8 = 100 e^200r models the rate of decay....divide by 100 then ln to get

 

ln .08 = 200 r

-.012628643 = r

 

So the general formula for this problem is 

A = Pe^{-.012628643 t}

 

So half life means when would you have half the mateiral

 

So if you started withh 100, when is it 50?

 

50 = 100e^{-.012628643 t    divide by 100 and ln}

ln .5 = -.012628643 t then divide to get

54.886 = t

 

So the half life of the substance is approximately 55 days.

 

Hope this helped