Question

Suppose that you take 200 mg of an antibiotic every 8 hr. The​ half-life of the drug is 8 hr​ (the time it takes for half of the drug to be eliminated from your​ blood). Use infinite series to find the​ long-term (steady-state) amount of antibiotic in your blood exactly.

Answer 1

Answer:

600 mg

Explanation:

The initial amount of the drug = 200 mg

The half-life of the drug = 8 hrs

It means that:-

After 6 hours, the concentration becomes :- $\frac{200}{2}$ mg

After 12 hours, the concentration becomes :- $\frac{200}{4}$ mg

After 18 hours, the concentration becomes :- $\frac{200}{8}$ mg

And so on...

Thus,

After infinite time = $200+\frac{200}{2}+\frac{200}{4}+\frac{200}{8}+..$

Thus,

After infinite time = $200\times (1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+..)$

The sum of the infinite series is:- $1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+..$ = $\frac{1}{1+\frac{1}{2}}=2$

So,

After infinite time = $200\times 2$ mg = 600 mg