Advanced math question...please help?

Question

If a person takes a dose of a certain drug, research shows that the body will eliminate 25% of that drug from the bood stream in a four hour period. If a person takes a 16ml dose how long will it take to elminate the drug from the blood?

Answer 1

This is the stuff of mystery novels. In theory, it is never eliminated, the concentration just gets too small to measure.

The real questions are:

1) What is the minimum detectable concentration?
2) What is the "normal" concentration (if applicable)

If you know those numbers, you can set up an equation to determine when the drug is no longer detectable.

C = 16 * (0.75)**(t/4)

A spreadsheet will give you a rough idea:

0.0 16.00
4.0 12.00
8.0 9.00
12.0 6.75
16.0 5.06
20.0 3.80
24.0 2.84
28.0 2.13
32.0 1.60
36.0 1.20
40.0 0.90
44.0 0.68
48.0 0.51

Answer 2

Mathematically, it'll never happen. You're talking here about exponential decay. You could model it in many ways, but the easiest would be

C(t) = c (3/4)^(t/4)

where t is time, measured in hours, and c is the initial concentration. The limit of this as t gets large without bound is 0, but the equation itself will never reach 0.

It's important to recognize that this model is based on the assumption that you have a macroscopic sample of the drug in your bloodstream. Once the amount drops to a few molecules, the model no longer applies. Think about it: suppose there was one molecule in your body. Does that mean that, in four hours, there'd be 3/4 of a molecule there? Clearly not. At that point, about the best you can say is that "after four hours, there's a 1/4 chance that the molecule has been eliminated". Again, mathematically, the probability that that one molecule is still around approaches 0, but it will never get there.

So, the short version of the answer is: it is impossible to say when or if the drug will be completely eliminated.

Answer 3

In 4 hrs, there will be 12mL left. 4 hrs more will leave 9mL. 4 hrs more (we are at 12 hrs now) leaves 6.75. Another 4 has 5.0625 mL in the blood. And keep going...