If you take 500mg of Acetaminophen, how many hours will it take for it to be expelled from your body if it expels at a rate of 20% per hour?

Question

not sure if this is a half life question or an exponential decay question 

Answer 1

Theoretically, the amount remaining approaches zero, but is never zero for any finite time, so you have to pick a very small target amount that is greater than zero, and calculate the time it takes for the initial amount to drop to that small target amount.  A final target value of 1 mg, as in Mark's solution, is reasonable, since 1 mg is only 0.2% of the initial amount of 500 mg.  You can work this problem as exponential decay. 

 

A(t) = A₀e^-kt, k = hours^-1, positive

 

Get k from the data for t = 1 hr, when (100 - 20)% = 80% remains.

 

0.80(500) = 500e^-k(1)

 

k = -ln0.8 ≅ 0.22314

 

A(t) = 500 e^{-0.22314 t} 

 

Now pick a very small final A, like 1 mg.

 

1 = 500 e^{-0.22314 t} 

 

Solve for t and get

 

t = ln(1/500) / (-0.22314) ≅ 27.85 hours

 

So it takes about 27.85 hours for 500 mg to drop to 1 mg.

 

Note that Mark's solution gives the same time:

 

ln 0.002 > h (ln 0.8)

 

Since ln(0.8) is negative, 

 

h > ln(0.002) / ln(0.8)

 

h > 27.85 hours

 

So you have 2 methods of working this type of problem and you have sufficient information to figure out a "half-life" if you wish.

Answer 2

It is decay, not exponential - that would involve "e"

1 > 500(0.80)^h

0.002 > 0,80^h

ln 0.002 > h (ln 0.8)

 

Can you solve for h and answer?