Hydrocodone bitartrate is used as a cough suppressant. After the drug is fully absorbed, the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with constant of proportionality . Suppose that the half-life of hydrocodone bitartrate in the body is 3.9 hours, and that the oral dose taken is 11 mg.

(a) Write a differential equation for the quantity, , of hydrocodone bitartrate in the body at time , in hours, since the drug was fully absorbed.

To derive the differential equation, let's denote Q(t) as the quantity of hydrocodone bitartrate in the body at time t.

We know that the quantity of drug in the body decreases at a rate proportional to the amount left in the body, with the constant of proportionality being λ.

The half-life of hydrocodone bitartrate is the time it takes for the quantity to decrease to half its original value. In this case, we are given that the half-life is 3.9 hours.

The exponential decay formula gives us the relationship between the half-life and the constant of proportionality:

t1/2 = ln(2)/λ

Solving for λ, we have:

λ = ln(2)/t1/2

Substituting the given half-life of 3.9 hours:

λ = ln(2)/3.9

Now we can write the differential equation:

dQ/dt = -λ * Q

Substituting λ, we get:

dQ/dt = -(ln(2)/3.9) * Q

Therefore, the differential equation for the quantity of hydrocodone bitartrate in the body at time t is:

dQ/dt = -(ln(2)/3.9) * Q

To write the differential equation for the quantity of hydrocodone bitartrate in the body at time t, we can use the information that the quantity decreases at a rate proportional to the amount left in the body.

Let Q(t) represent the quantity of hydrocodone bitartrate in the body at time t. We know that the amount left in the body decreases at a rate proportional to Q(t). Let's denote the constant of proportionality as k.

Therefore, the differential equation can be written as:

dQ/dt = -k * Q(t)

where dQ/dt represents the rate of change of the quantity with respect to time.

Now, we need to find the value of k using the information about the half-life of hydrocodone bitartrate. The half-life is the time it takes for the quantity to reduce to half its initial value. In this case, the half-life is given as 3.9 hours.

We know that for a general first-order reaction, the half-life (t1/2) is related to the rate constant (k) as:

t1/2 = (ln(2))/k

Substituting the given value of t1/2 = 3.9 hours, we can solve for k:

3.9 = ln(2)/k

k = ln(2)/3.9

Therefore, the differential equation for the quantity of hydrocodone bitartrate in the body at time t is:

dQ/dt = - (ln(2)/3.9) * Q(t)