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A doctor injects you with 500 milligrams of a radioactive compound for medical imaging. It has a half-life of 6 hours. How long will it take for 99.9% of it to decay? Round to the nearest hour.

Answer

99.9*500 = 500e^kt

99.9 = e^kt

99.9 = e^6k

ln99.9 = lne^6k

4.6041 = 6k

k = 0.76736

I actually made some progress. Now I'm stuck here. How do i proceed?

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    $\begingroup$ Welcome to Mathematics SE! have you considered how to use the half live you've been given? $\endgroup$ Commented Oct 14, 2019 at 9:52
  • $\begingroup$ 99.9*500 = 500e^kt 99.9 = e^kt 99.9 = e^6k ln99.9 = lne^6k 4.6041 = 6k k = 0.76736 I actually made some progress. Now I'm stuck here. How do i proceed? $\endgroup$
    – sirkal
    Commented Oct 14, 2019 at 10:00
  • $\begingroup$ something else you should consider is if the amount of radioactive substance effects the time taken for a given % amount to decay $\endgroup$ Commented Oct 14, 2019 at 10:04
  • $\begingroup$ There are some problems with your solution. First of all, rethink what the problem statement means by "99.9% of it to decay". How much is left after that? Secondly, please review the definition of half-life and it's possible relation with the number $e$. Here's a similar question:math.stackexchange.com/questions/3337801/… $\endgroup$
    – Matti P.
    Commented Oct 14, 2019 at 10:17

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Always divide percentages by $100$ to get proportions in contexts like this. If $99.9\%$ (a proportion of $0.999$) has decayed, you're left with $0.001$ times the original total. Let's work in hours. Since $e^{-6k}=\frac12$, $e^{-kt}=0.001\implies t=6\frac{\ln 1000}{\ln 2}\approx60$.

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  • $\begingroup$ Thank you for the answer. I think I now understand where I was going wrong $\endgroup$
    – sirkal
    Commented Oct 14, 2019 at 10:29

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