suppose m(0) milligrams of a drug are put in the blood of an injection. The amount of drug t hours after the injection is given by
m(t)=m(o)e^-kt, for t (=>) 0, where k is the rate constant, which is related to the half life. we also treat oral administration
of drugs as an injection, although the model is less accurate because of the drug must be absorbed into the blood through the stomach
4.
the sedative diazepam has a half-life of 7 hr. find the drug function m for diazepam.
graph the drug function with m(0)=5 for 0(<=) t (<=)48 hours.
how much drug remains in the blood 12 hr and 24 hr after a 5-mg dose is taken?
5.
the antibiotic tetracycline has a half-life of 9 hours. suppose a doctor wishes a patient to have a 100mg of tetracycline in the blood 18 hours after an injection. what initial does meets his requirement?
6.
twelvee hours after a 200 mg dose of a drug is injected. the drug level in thebloodd is 75mg. what is the approximate half-life drug?
To find the drug function for diazepam, we can use the given formula:
m(t) = m(0) * e^(-kt)
1. Find the drug function m for diazepam:
Since the half-life of diazepam is 7 hours, we can use this information to find the rate constant k.
The half-life of a substance is the time it takes for half of it to decay or disappear. In this case, after 7 hours, the amount of drug remaining will be half of the initial dose.
Given:
Half-life of diazepam = 7 hours
Initial dose m(0) = 5 mg
To find k, we can rearrange the formula as follows:
1/2 = e^(-k * 7)
Taking the natural logarithm of both sides to solve for k:
ln(1/2) = -k * 7
k = ln(1/2) / -7
Once we have the value of k, we can substitute it into the drug function equation:
m(t) = 5 * e^(-kt)
2. Graphing the drug function:
Now we can graph the drug function for diazepam with m(0) = 5 and t ranging from 0 to 48 hours. To do this, plot the values of m(t) for different values of t within this range.
3. Finding drug levels at 12 and 24 hours:
To find the amount of drug remaining in the blood 12 and 24 hours after a 5 mg dose is taken, we substitute those values of t into the drug function equation and calculate m(t).
for t=12:
m(12) = 5 * e^(-k * 12)
for t=24:
m(24) = 5 * e^(-k * 24)
4. Finding the initial dose for meeting the requirement:
We are given that the desired drug level in the blood 18 hours after injection is 100 mg for tetracycline. To find the initial dose that meets this requirement, we need to solve for m(0) in the drug function equation:
100 = m(0) * e^(-k * 18)
Solving for m(0) gives us the initial dose required.
5. Approximating the half-life of a drug:
If the drug level in the blood decreases from 200 mg to 75 mg after 12 hours, we can use this information to find the approximate half-life of the drug. We need to determine the time it takes for the drug level to decrease to half of the initial dose (200 mg/2 = 100 mg). The time it takes for this decrease will be approximately equal to the half-life of the drug.