# Math Help - Exponential Decay-Radioactivity

1. ## Exponential Decay-Radioactivity

A sample of a radioactive substance decayed to 95.5% of its original amount after a year. What is the half-life of the substance in years, and how long would it take the sample to decay to 25% of its original amount in years.

My roommate just asked me this and I don't remember how to do it! Please help!

2. Originally Posted by acg716
A sample of a radioactive substance decayed to 95.5% of its original amount after a year. What is the half-life of the substance in years, and how long would it take the sample to decay to 25% of its original amount in years.

My roommate just asked me this and I don't remember how to do it! Please help!
Let $N_0$ be the original amount of the substance and $N$ be the current amount of the substance.

Now, radioactive decay takes on the form $N=N_0e^{-kt}$, where $k$ is the decay constant. We need to know what $k$ is, because the half life of this substance is defined by $\lambda=\frac{\ln 2}{k}$, where $\lambda$ is the half life.

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The first bit of information gives us enough info to find k.

It takes a year for the substance to decay to 95.5% of the original amount.

Thus, we see that $N=N_0e^{-kt}\implies .955N_0=N_0e^{-k}$, where we replaced $N$ with $.955N_0$.

This now simplifies to $.955=e^{-kt}\implies \ln(.955)=-k\implies \color{red}\boxed{k\approx 0.046}$

Now we can find the half life:

$\lambda=\frac{\ln 2}{0.046}\implies\color{red}\boxed{\lambda\approx 15.05~\text{years}}$

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Now, can you find $t$, such that $.25=e^{-0.046t}$?? [this was what the second part of the question was asking for]

Does this make sense?

--Chris

3. thank you so much!!