Two 5-grain aspirin tablets contain 650 mg of the drug. With aspirin's half-life of 29 minutes, how much is left in the bloodstream after 2 hours? How long does it take for the level to be equivalent to 10 mg of aspirin? If an individual takes two tablets every 4 hours, what is the maintenance level of the aspirin?

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To answer these questions, we need to understand the concept of half-life and how it relates to the dosage of aspirin.

First, let's calculate the initial dosage of aspirin in the bloodstream. Two 5-grain aspirin tablets contain a total of 2 x 5 = 10 grains of aspirin.

Since the dosage is given in grains, we need to convert it to milligrams (mg) for consistency. 1 grain is approximately equal to 65 mg, so 10 grains of aspirin would be 10 x 65 = 650 mg.

Now, let's calculate how much aspirin is left in the bloodstream after 2 hours.

The half-life of aspirin is given as 29 minutes. This means that every 29 minutes, the amount of aspirin in the bloodstream is reduced by half.

To calculate the remaining amount of aspirin after 2 hours (120 minutes), we can divide the time by the half-life and raise 0.5 to the power of the result.

Remaining amount = Initial amount x (0.5)^(time / half-life)

In our case, the initial amount is 650 mg, the time is 120 minutes, and the half-life is 29 minutes.

Remaining amount = 650 x (0.5)^(120 / 29) ≈ 650 x 0.058 ≈ 37.7 mg

So, after 2 hours, approximately 37.7 mg of aspirin is left in the bloodstream.

Next, let's determine how long it takes for the level of aspirin to reach 10 mg.

Using the same formula as before, we can solve for time:

Remaining amount = Initial amount x (0.5)^(time / half-life)

Substituting the values, we have:

10 = 650 x (0.5)^(time / 29)

Dividing both sides by 650, we get:

0.0154 = (0.5)^(time / 29)

Taking the logarithm base 0.5 of both sides:

log(0.5)(0.0154) = time / 29

Simplifying and solving for time, we find:

time ≈ log(0.5)(0.0154) x 29 ≈ -6.7 x 29 ≈ -194.3 minutes

Keep in mind that the half-life equation assumes an exponential decay, so a negative time value does not make sense in this context. Therefore, it indicates that it would take more than 194 minutes for the level of aspirin to reach 10 mg, which is not possible.

Lastly, let's determine the maintenance level of aspirin if an individual takes two tablets every 4 hours.

With a half-life of 29 minutes, taking two tablets every 4 hours helps maintain a relatively stable level of aspirin in the bloodstream.

To calculate the maintenance level, we need to account for the time it takes for the previous dose to be metabolized. Since the half-life is 29 minutes, let's calculate how many dosages can be taken in 4 hours:

Time in minutes = 4 hours x 60 minutes/hour = 240 minutes
Number of doses = Time in minutes / half-life = 240 / 29 ≈ 8.28 (approximately)

Since it is not possible to take a fraction of a dose, it is best to take the lower integer value, which is 8 doses.

Maintenance level = Initial dosage x (0.5)^(number of doses)

Substituting the values, we have:

Maintenance level = 650 mg x (0.5)^8 ≈ 19.5 mg

Thus, the maintenance level of aspirin, if an individual takes two tablets every 4 hours, is approximately 19.5 mg.