SOLUTION: Paracetamol is a drug used to treat mild to moderate pain and to reduce fever. It has a half-life in the body of 3 hours. Under a Doctor's care, patient N is given a dose of a cert

Algebra ->  Exponential-and-logarithmic-functions -> SOLUTION: Paracetamol is a drug used to treat mild to moderate pain and to reduce fever. It has a half-life in the body of 3 hours. Under a Doctor's care, patient N is given a dose of a cert      Log On


   

Question 1171696: Paracetamol is a drug used to treat mild to moderate pain and to reduce fever. It has a half-life in the body of 3 hours. Under a Doctor's care, patient N is given a dose of a certain paracetamol. Patient N asks the doctor for another same dose of this paracetamol. The doctor agrees to allow patient N to take dose once there is only 6.25% of the original dose remaining in the body. How long will patient N have to wait, before taking another dose of this paracetamol?
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342) About Me  (Show Source):
You can put this solution on YOUR website!
formula for half life is:
1/2 = (1 + r) ^ n
when n = 3, the formula becomes:
1/2 = (1 + r) ^ 3
solve for r to get:
r = (1/2) ^ (1/3) - 1 = -.206299474.
that's your growth rate per time period.
confirm by replacing r in the original equation to get:
1/2 = (1 + r) ^ 3 becomes:
1/2 = (1 - .206299474) ^ 3 = 1/2
this confirms the value of r is good.

to find how long the patient has to wait for there to be only 6.25% of the original dose in his system, the the formula becomes.
.0625 = 1 * (1 - .206299474) ^ n
simplify to get:
.0625 = (1 - .206299474) ^ n
take the log of both sides of the equation to get:
log(.0625) = log((1-.206299474)^n)
since log(x^n) = n * log(x), this becomes:
log(.0625) = n * log(1 - .206299474)
solve for n to get:
n = log(.0625) / log(1 - .206299474) = 12.
confirm by replacing n in the original equation to get:
.0625 = (1 - .206299474) ^ n becomes:
.0625 = (1 - .206299474) ^ 12 = .0625
this confirms the value of n is good.

your solution is that the patient will have to wait 12 hours before taking another dose.



Answer by ikleyn(52803) About Me  (Show Source):
You can put this solution on YOUR website!
.
Paracetamol is a drug used to treat mild to moderate pain and to reduce fever. It has a half-life in the body of 3 hours.
Under a Doctor's care, patient N is given a dose of a certain paracetamol. Patient N asks the doctor for another same dose
of this paracetamol. The doctor agrees to allow patient N to take dose once there is only 6.25% of the original dose
remaining in the body. How long will patient N have to wait, before taking another dose of this paracetamol?
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Notice that  6.25%  is this ratio   6.25%2F100 = 1%2F16 = 1%2F2%5E4.

It means that after  4  (four)  half-life periods the amount of the drug in the body will reach  6.25%  of the original level.

So,  the  ANSWER  to the problem's question is  4  times half-life period,  or  4 x 3 hours = 12 hours.

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