# Pre Calculus

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When doctors prescribe medicine at equally spaced time intervals, they are aware that the body metabolizes the drug gradually. After some period of time, only a certain percent of the original amount remains. After each dose, the amount of the drug in the body is equal to the amount of the given dose plus the amount remaining from the previous doses. The amount of the drug present in the body after nth doses is modelled by a geometric series where t1 is the prescribed dosage and r is the previous dose remaining in the body.
Suppose a person with an ear infection takes a 200-mg ampicillin tablet every 4 h. About 12% of the drug in the body at the start of a four-hour period is still present at the end of that period. What amount of ampicillin is in the body, to the nearest tenth of a milligram,
(a) after taking third tablet?
(b) after taking the sixth tablet?

• Pre Calculus -

amount right now (1st pill) = 200 mg
amount after 2nd pill = 200 + .12(200) = 200(1.12)
amount after 3rd pill = 200 + 200(.12) + 200(.12)^2

amount after 4th pill = 200 + 200(.12) + 200(.12)62 + 200(.12)^3

...

amount after nth pill = 200 + 200(.12) + ... + 200(.12)^(n-1)

after 6th pill
amount = 200 + 200(.12) + ... + 200(.12)^5
= 200( 1 - .12^6)/(1-.12) = 227.27 mg

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