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?
4. To find the drug function m(t) for diazepam, we can use the given information that the half-life of diazepam is 7 hours. The half-life is the time it takes for the amount of drug to reduce to half its initial value.
In the equation m(t) = m(0)e^(-kt), we need to find the value of k. Since the half-life is 7 hours, we know that after 7 hours, the amount of drug remaining will be half of the initial amount.
So, we can substitute t = 7 and m(t) = (1/2)m(0) into the equation and solve for k:
(1/2)m(0) = m(0)e^(-7k)
Dividing both sides by m(0), we get:
1/2 = e^(-7k)
To solve for k, take the natural logarithm (ln) of both sides:
ln(1/2) = -7k
Now, we can solve for k:
k = ln(1/2) / -7
With the value of k, we can now find the drug function m(t). For m(0) = 5, we have:
m(t) = 5e^(-kt)
Now, to graph the drug function with m(0) = 5 for 0 ≤ t ≤ 48 hours, you can plot points using different values of t within the given range and evaluate m(t) using the equation.
To find how much drug remains in the blood 12 hours and 24 hours after a 5-mg dose is taken, substitute the respective values of t into the drug function m(t) and evaluate the result.
5. To find the initial dose of tetracycline that meets the requirement of having 100mg in the blood after 18 hours, we can use the drug function equation with the given half-life of 9 hours.
Using m(t) = m(0)e^(-kt), we need to find the value of m(0).
Substituting t = 18, m(t) = 100, and k = ln(1/2) / -9, we have:
100 = m(0)e^(-9 * 18)
Simplifying, we get:
100 = m(0)e^(-162)
Dividing both sides by e^(-162), we get:
m(0) = 100 / e^(-162)
To find the approximate initial dose, evaluate m(0) using a calculator.
6. To find the approximate half-life of the drug given that 12 hours after a 200mg dose, the drug level is 75mg, we can use the drug function equation.
Using m(t) = m(0)e^(-kt), we need to find the value of k, which is related to the half-life.
Substituting t = 12, m(t) = 75, and m(0) = 200 into the equation, we have:
75 = 200e^(-k * 12)
Dividing by 200, we get:
0.375 = e^(-12k)
Taking the natural logarithm of both sides:
ln(0.375) = -12k
Now, solve for k:
k = ln(0.375) / -12
With the value of k, we can find the half-life by taking the reciprocal:
half-life = 1 / k
Evaluate half-life using a calculator to get the approximate value.