word problem

Question

The half-life of a certian medication in the bloodstream is 36 hours. If you take a dosage of 100 mg, in how may hours will there be 40 mg of the medication in your bloodstream? Round to the nearest hour.

 

Please show work

Answer 1

After 36 hrs 50mg remain; after 72 hrs 25mg remain; after 108 hrs 12.5 mg remain. Therefore, 2^-(1/36) is the rate of dissolution per hour. The amount in milligrams after time t hrs is x=100*2^-(t/36). Let's see how this works. At t=36 x=100*2^-1=100/2=50; t=72, x=100*2^-2=100/4=25; t=108 x=100*2^-3=12.5.

So if x=40, 100*2^-(t/36)=40, 2^-(t/36)=0.4. Take logs: -tlog(2)/36=log(0.4)=-0.39794; t=-36log(0.4)/log(2)=47.59 hrs. So the time to reach 40mg is 48 hours, to the nearest hour.