SOLUTION: A certain pain reliever has a half-life of 16 hours. If the initial plasma level of this pain reliever, given as a single dose, is 512mg/L, how long will it take for the plasma lev

Question 1171697: A certain pain reliever has a half-life of 16 hours. If the initial plasma level of this pain reliever, given as a single dose, is 512mg/L, how long will it take for the plasma level to fall to 16mg/L?
Found 3 solutions by Theo, ikleyn, MathTherapy:
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
with discrete compounding, the formula to use is:
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period.
n is the number of time periods.

when you know what f is and what p is and what n is, you need to find r.
the formula becomes:
1 = 2 * (1 + r) ^ 16
divide both sides of the equation by 2 to get:
1/2 = (1 + r) ^ 16
take the 16th root of both sides of the equation to get:
(1/2) ^ (1/16) = 1 + r
solve for 1 + r to get:
1 + r = .9576032807.
solve for r to get:
r = .9576032807 - 1 = -.0423967193.
that's your growth rate per time period.
if it's positive, you're growing.
if it's negative, you're shrinking.
in this case, you're shrinking.

to see if the growth rate is good, replace r in the equation by the growth rate to get:
1 = 2 * (1 - .0423967193) ^ 16
solve to get:
1 = 1, confirming the value of r is good.

when f = 16 and p = 512, the formula becomes:
16 = 512 * (1 - .0423967193) ^ n
divide both sides of this formula to get:
16 / 512 = (1 - .0423967193) ^ n
take the log of both sides of this equation to get:
log(16/512) = log((1-.0423967193)^n)
since log(x^n) = n * log(x), this equation becomes:
log(16/512) = n * log(1 - .0423967193)
solve for n to get:
n = log(16/512) / log(1 - .0423967193) = 80.
that's your solution.

confirm by replacing n in the original equation to get:
16 = 512 * (1 - .0423967193) ^ n becomes:
16 = 512 * (1 - .0423967193) ^ 80 which becomes:
16 = 16.
this confirms the solution is correct.

the solution is that that the plasma level of the pain reliever will fall from 512 milligrams per liter to 16 milligrams per liter in 80 hours.

Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
A certain pain reliever has a half-life of 16 hours. If the initial plasma level of this pain reliever,
given as a single dose, is 512mg/L, how long will it take for the plasma level to fall to 16mg/L?
~~~~~~~~~~~~~~~

Notice that the ratio of 16 mg/L to 512 mg/L is    = = .

It means that after 5 (five) half-life periods the original concentration
of the drug in the plasma will decrease from the original 512 mg/L to 16 mg/L.

So, the ANSWER to the problem's question is 5 times half-life periods, or 5 x 16 hours = 80 hours.

You do not need to make long unnecessary calculations to answer the problem's question.

Answer by MathTherapy(10552)   (Show Source): You can put this solution on YOUR website!

A certain pain reliever has a half-life of 16 hours. If the initial plasma level of this pain reliever, given as a single dose, is 512mg/L, how long will it take for the plasma level to fall to 16mg/L?

It's not nearly as complex as the other person makes it out to be.
With ½-life being 16 hours, DECAY CONSTANT, or
The CONTINUOUS GROWTH/DECAY formula: , then becomes:
------- Converting to LOGARITHMIC (Natural) form

Time taken for the plasma level to fall to 16 mg/L, or
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