Drug formulation designed to release a drug into the body at a predetermined rate over a prolonged period of time
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"Time-release" drugs have the advantage of releasing the drug to the body at a constant rate so that the drug concentration at any time is not too high as to have harmful side effects or too low as to be ineffective. A schematic diagram of a pill that works on this basis is shown. Explain how it works.
Drug Dosage Certain medical conditions are treated with a fixed dose of a drug administered at regular intervals. Suppose a person is given 2 mg of a drug each day and that during cach 24 -hr period, the body utilizes $40 \%$ of the amount of drug that was present at the beginning of the period. (a) Show that the amount of the drug present in the body at the end of $n$ days is $$ \sum_{i=1}^{n} 2(0.6)^{i} $$ (b) What will be the approximate quantity of the drug in the body at the end of each day after the treatment has been administered for a long period of time?
Further Topics in Algebra
Geometric Sequences and Series
Many drugs are eliminated from the body in an exponential manner. Thus, if a drug is given in dosages of size $C$ at time intervals of length $t,$ the amount $A_{n}$ of the drug in the body just after the $(n+1)$ st dose is $$A_{n}=C+C e^{-k t}+C e^{-2 k t}+\cdots+C e^{-n k t}$$ where $k$ is a positive constant that depends on the type of drug. (a) Derive a formula for $A$, the amount of drug in the body just after a dose, if a person has been on the drug for a very long time (assume an infinitely long time). (b) Evaluate $A$ if it is known that one-half of a dose is eliminated from the body in 6 hours and doses of size 2 milligrams are given every 12 hours.
Infinite Series
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