Steadily-reducing drug dose question.?

Question

You wouldn't believe that I've got a degree in maths, but I can't answer this. My friend, the local pharmacist asked me; this guy is given 20 grams of drug in 100 mls. of water. He's instructed to take 5 mls each day of the solution and top the bottle of solution up with H2O. So the first day he gets 1 gram of drug. The next day, when he takes his 5 mls. of solution, it only contains 0.95 grams of drug, because he's already consumed some of the drug and diluted it. And so on. What is the formula that tells you how many grams of the drug he gets on the nth day?

Answer 1

20 in 100
19 in 100
18.05 in 100

Answer 2

zee_prime,
Pardon the pun but this has an "e-zee compounding solution." The amount of drug in the 5ml after n days is simply:

(19/20)^(n-1) grams = (0.95)^(n-1) grams

first day = (0.95)^0 = 1 gram
2nd (0.95)^1 = 0.95 grams
10th day = (0.95)^9 = 0.63 grams
100th day = 0.00623 grams

It's easy to see that this must be the case since when the 5 ml of water is added to the 95ml to top it off at 100ml then you automatically dilute the solution by a factor of 19/20 = 0.95

Answer 3

Exponential decay. First order differential equation, initial value problem

y = amout of drug in the bottle at any time

y/(100) = concentration of drug at any time,

dy/dt = the rate of change of the amount of drug in the bottle with respect to time

dy/dt = amount drug/time coming into the bottle - amount of drug/time going out of the bottle

amount of drug/time coming out = concentration of drug x amount of diluted solution coming out/time

amount of drug/time coming in = 0, no concentration of drug, pure water.

dy/dt = 0 - (y/100)*5

dy/dt = -y/20

dy/y = -dt/20

ln y = -t/20 + ln C, choice of constant in ln form

y/C = exp (-t/20)

y = C exp (-t/20)

To determine C for this particular problem, at t=0, y=20.
So C = 20 grams the initial amount of drug.

y = 20 exp (-t/20)

is the equation that is tailored for your drug problem. :)

Practically at the 20th day the amount of concentration in the bottle is 1/3 of the original concentration. By the 40th day, only about 1/14. By the 100th day less that 1/100. By this trend, I think continued medication after the 20th day is like taking pacebo. Our bodies repond only by half if the stimuli is decreased ten times. (logarithmic response to stimuli)