Medicine half life
A short challenge concerning the decay of medicines in the body.
Problem
A certain medicine is broken down by the body so that half of the remaining dose in the system is broken down every 12 hours.
If I regularly take 400mg of medicine in the morning and 400mg of medicine in the evening what is the long term effective level in my system after I take each tablet?
Extension: What will be the long term stable level after I take each tablet if the body breaks down half of the remaining dose every 6 hours?
If I regularly take 400mg of medicine in the morning and 400mg of medicine in the evening what is the long term effective level in my system after I take each tablet?
Extension: What will be the long term stable level after I take each tablet if the body breaks down half of the remaining dose every 6 hours?
Did you know ... ?
The 'half-life' of medicines is an important concept in medicine, particularly in the area of anti-depressants which take some time to reach active levels in the body.
The 'half-life' of medicines is an important concept in medicine, particularly in the area of anti-depressants which take some time to reach active levels in the body.
Student Solutions
As the question indicates a long-term level I might imagine having taken the tablets for several days and will assume that the tablets are taken in 12-hourly intervals. When I take a tablet, the amount remaining of the previous tablet will have halved, the tablet before that quartered an so on.
If I think carefully and clearly about this then I can write down an expression for the total mass $M(n)$ remaining in the system upon taking the $n$th tablet:
(With such an expression I should check that the limits are correct: My formula gives$M(1) = 400(1)$ and $M(2) = 400( 1+\frac{1}{2^1})$, which are correct).
$M(n)$ is a geometric series, which sums to
The long term level is found by taking the large $n$ limit, giving a level of $800$mg in the system, as one might intuitively guess.
Extension:
If the body breaks down half of the dose every $6$ hours then the levels of the previous tablets will have quartered each time. Thus the relevent geometric series is
The sum of this series is
The long term level will be $533\frac{1}{3}$mg