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.

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?

• Welcome to Mathematics SE! have you considered how to use the half live you've been given? Commented Oct 14, 2019 at 9:52
• 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? Commented Oct 14, 2019 at 10:00
• something else you should consider is if the amount of radioactive substance effects the time taken for a given % amount to decay Commented Oct 14, 2019 at 10:04
• 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/… Commented Oct 14, 2019 at 10:17

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 $$e^{-6k}=\frac12$$, $$e^{-kt}=0.001\implies t=6\frac{\ln 1000}{\ln 2}\approx60$$.