November 17th, 2017, 05:49 PM  #1 
Newbie Joined: Nov 2017 From: United States Posts: 1 Thanks: 0  How do I find the halflife
In 2 years, 20% of a radioactive element decays. Find its halflife rounded to 2 decimal places. I think k=LN(0.20)/2, but I'm unable to figure out how to solve for the halflife. 
November 17th, 2017, 05:57 PM  #2 
Math Team Joined: Jul 2011 From: Texas Posts: 2,678 Thanks: 1339 
$0.8 = \left(\dfrac{1}{2}\right)^{2/h}$ solve for the halflife, $h$ 
November 17th, 2017, 06:42 PM  #3 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,875 Thanks: 766 
I prefer to write any problem involving "half life" as powers of 1/2. Since skeeter beat me to that, here is how to do it in a more "formulaic" method. Any such problem (constant rate of decay) can be written in the form $S(t)= Ce^{kt}$. When $t = 0$, $S(0) = C$, the initial amount. When $t = 2$, $S(2) = Ce^{2k}$ and we are told that is $C  0.2C = 0.8C$, so $e^{2k}= 0.8$. Hence $2k= \ln(0.8)$ or $k= \frac{\ln(0.8)}{2}$. Hence the formula is $\displaystyle S(t)= Ce^{\frac{\ln(0.8)t}{2}}= C\left(e^{\ln(0.8)}\right)^{t/2}= C(0.8^{t/2})$. "Half life" is the value of $t$ that makes that $C/2$. We need to solve $C(0.8^{t/2})= C/2$ or $0.8^{t/2}= \frac{1}{2}$. That is the same as skeeter's solution. Last edited by skipjack; November 17th, 2017 at 08:09 PM. 

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