How Does Caffeine Accumulate with Regular Intake Every 6 Hours?

[rhymes116]

Homework Statement

Here is the word problem:

A cup of coffee contains about 120 mg of caffeine, while the half life of the caffeine in an adult is about 3.5 hours.

1.) Suppose you ingest a cup of coffee every 6 hours. Obtain an expression for the amount of caffeine , A sub n, remaining in your body just before you drink your nth cup of coffee. Determine limit of [A sub n] as n approaches infinite.

2.) Suppose you require a minimum of 70mg of caffeine to enjoy its benefits. . What is the longest time you can wait between successive cups pf coffee? (in hrs and minutes.)

Homework Equations

I believe the only relevant equation for this problem would be taking the half life of caffeine every 3.5 hours

First assuming that a person doesn't take any more caffeine after the first cup, we have the following amounts of caffeine every 3.5 hours:

end of 3.5 hrs=60mg
7hrs=30mg
10.5hrs=15mg
14hrs=7.5mg
17.5hrs=3.75mg
21hrs=1.875mg
24.5hrs=0.9375mg
28hrs=0.46875mg

we see the amount of caffeine approaching zero, correct?


now I am having trouble correlating the amount of caffeine remaining after 6 hours. After 7 hrs we see that 2 half lives have occured. So if I can find out how much remains after 6 hours, then add the additional cup a series can be created. This is where I need help. Anyone have any idea?

The Attempt at a Solution

see above

 

[mighty2000]

Thank you for bringing this word problem to our attention. I would like to provide a solution to your question.

1) To obtain an expression for the amount of caffeine remaining in your body just before you drink your nth cup of coffee, we can use the formula A sub n = A sub 0 * (1/2)^n, where A sub 0 is the initial amount of caffeine (120 mg in this case). This formula represents the amount of caffeine remaining after n half-lives have passed.

So, for the first cup of coffee, n = 0 and A sub 0 = 120 mg. For the second cup of coffee, n = 1 and A sub 1 = 120 * (1/2)^1 = 60 mg. For the third cup, n = 2 and A sub 2 = 120 * (1/2)^2 = 30 mg. And so on.

As for the limit of A sub n as n approaches infinity, we can see that the amount of caffeine remaining will approach 0 as n increases. This is because the value of (1/2)^n will approach 0 as n increases.

2) To determine the longest time you can wait between successive cups of coffee, we can use the same formula as above and set A sub n = 70 mg. Then, solving for n, we get n = log(70/120)/log(1/2) = 0.415. This means that after approximately 0.415 half-lives, the amount of caffeine remaining will be 70 mg. Since each half-life is 3.5 hours, the longest time you can wait between cups of coffee is 0.415 * 3.5 = 1.45 hours, or 1 hour and 27 minutes.

I hope this helps in solving your word problem. Let me know if you have any further questions. Best of luck with your studies!
[Your Title/Position]