Question 1179243: A drug is administered to a patient every 4 hours. The patients metabolism causes the concentration of the drug in the patients blood to decrease by 75% between doses. The dose causes an instantaneous rise in blood concentration of 0.2 units. Then what will be the concentration of drug in the patients blood immediately after the nth dose
Thank you for your help!!
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! as far as i can tell, the formula will be:
An = .2 * (the sum of .25^(n-x), from x = 1 to x = n)
for example:
when n = 2, the formula becomes:
A2 = .2 * (.25^1 + .25^0) = .25
when n = 3, the formula becomes:
A3 = .2 * (.25^2 + .25^1 + .25^0) = .2625
when n = 4, the formula becomes:
A4 = .2 * (.25^3 + .25^2 + .25^1 + .25^0) = .265625
the dose stabilizes at n = 7 when rounded to 4 decimal places.
the maximum amount of drug in the patient's system is .2667 when n = 7
it remains at .2667 from there on.
the following spreadsheet shows you the progression.
per the formula, when n = 7, the amount of drug in the patient's system will be:
A7 = .2 * sum(.25^(9-x) from x = 1 to x = 7) which becomes:
A7 = .2 * (.25^6 + .25^5 + .25^4 + .25^3 + .25^2 + .25^1 + .25^0) which becomes:
A7 = .2666503906
round to 4 decimal places and you get A7 = .26667.
the following spreadsheet does the calculations for each n.
for example:
when n = 1, h = 0 and A1 = .2 = .2000
when n = 2, h = 4 and A2 = .25 * A1 + .2 = .25 * .2 + .2 = .25 = .2500
when n = 3, h = 8 and A3 = .25 * A2 + .2 = .25 * .25 + .2 = .2625 = .2625
when n = 4, h = 12 and A4 = .25 * A3 + .2 = .25 * .2625 + .2 = .265625 = .2656
when n = 5, h = 16 and A5 = .25 * A4 + .2 = .25 * .265625 + .2 = .26640625 = .2664
when n = 6, h = 20 and A6 = .25 * A5 + .2 = .25 * .26640625 + .2 = .2666015625 = .2666
when n = 7, h = 24 and A7 = .25 * A6 + .2 = .25 * .2666015625 + .2 = .2666503906 = .2667
as you can see from the spreadsheet, the value remains at .2667 from there on when rounded to 4 decimal places.
here's the spreadsheet calculation for n = 1 to 112.
you can see that the amount of drugs in the patient's system stabilizes at n = 7 when the maximum amount in the patient's system is .2667 and remains at .2667 from there on.
you were asked what will be the concentration of drug in the patient's blood immediately after the nth dose.
that will be .2 * (the sum of .25^(n-x), from x = 1 to x = n), as best i can determine.
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