Radioactive Radium has a half-life of approximately 1600 years. What percentage of the present amount remains after 100 years?
Half-Life Exponential Decay using base e?
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exponential-function
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0Welcome to math.SE: since you are new, I wanted to let you know a few things about the site. In order to get the best possible answers, it is helpful if you say in what context you encountered the problem, and what your thoughts on it are; this will prevent people from telling you things you already know, and help them give their answers at the right level. – 2012-10-30
1 Answers
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The formula for exponential decay is: $\frac{dN}{dt}=-N\lambda$
Solving the differential equation, we get the following:
$N(t)=N(0)\cdot \rm{e}^{-\lambda t}$
To get the percentage, we will start off with $N(0)=100$, solving for $t=100$ and $\lambda=\frac{\ln{2}}{1600}$ (which we find from the definition of half-life: $t_{\frac{1}{2}}=\frac{\ln{2}}{\lambda}$), we get:
$N(100)=100\cdot\rm{e}^{-\frac{100\ln{2}}{1600}}=95.76$
So $95.76\%$ remains after $100$ years.
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0Actually: probably for someone asking this question, solving the differential equation would be out of his depth, and you would start with your second display, which is probably in his textbook. But your solution goes beyond that to benefit more advanced readers as well. Which is good. – 2012-10-30