A patient is given a drug intravenously at a rate of 43.2 mg/hour. The drug enters a compartment volume or the volume of the part of the body of 35,000 ml. The rate at which the drug leaves the body

(mg/hour) is proportional to the quantity present, with proportionality constant 0.082 per hour. The patient doesnt have any drugs initially.
(a) Describe in words how you would expect the concentration of the drug in the patient to vary with time.
(b)Write a differential equation satisfied by the concentration of the drug, c(t).
(c)Solve the differentail equation.

I get dw/dt=36.8-0.048D as my equation and when i solve i get D=Aexp(-.048)=36.8

am I off any help would be greatly appreciated.

They want c(t), which represents the concentration present. Concentration is mass present divided by volume (with units of mg/ml). Let time t be in hours.

dc/dt = (1/35,000)* [43.2 -35,000 c*0.082)
= (1/35,000)*[43.2 - 2870 c)

The steady state concentration (after very large t) is obtained when dc/dt= 0, and corresponds to
c = 0.015 mg/ml

(a) The drug concentration in the patient's body is expected to increase initially as the drug is being given intravenously at a constant rate. However, as time progresses, the drug concentration will reach a steady state because the rate at which the drug leaves the body is proportional to the quantity present.

(b) To derive the differential equation satisfied by the concentration of the drug, c(t), we can consider the fact that the rate of change of the drug concentration in the compartment is equal to the rate at which the drug enters minus the rate at which the drug leaves.

Let c(t) be the concentration of the drug in mg/mL at time t. The rate at which the drug enters the compartment is given as 43.2 mg/hour. The rate at which the drug leaves the body is proportional to the quantity present with a proportionality constant of 0.082 per hour. Therefore, the rate at which the drug leaves is 0.082 * c(t) mg/hour.

The differential equation satisfied by the concentration of the drug, c(t), is:

dc(t)/dt = 43.2 - 0.082 * c(t)

(c) To solve the differential equation, we can rearrange it as:

dc(t) = (43.2 - 0.082 * c(t)) dt

Separating variables and integrating both sides, we get:

∫(1 / (43.2 - 0.082 * c(t))) dc(t) = ∫dt

To solve this integral, we can use a substitution. Let u = 43.2 - 0.082 * c(t). Then du = -0.082 * dc(t).

Substituting these values, the integral becomes:

∫(-1 / 0.082) * (1 / u) du = ∫dt

Simplifying, we get:

-1 / 0.082 * ln|u| = t + C

where C is the constant of integration.

Substituting back, we have:

-1 / 0.082 * ln|43.2 - 0.082 * c(t)| = t + C

Solving for c(t), we get:

ln|43.2 - 0.082 * c(t)| = -0.082 * 0.082 * t - C * 0.082

Taking the exponential of both sides:

|43.2 - 0.082 * c(t)| = e^(-0.082 * 0.082 * t - C * 0.082)

Removing the absolute value, we have:

43.2 - 0.082 * c(t) = ± e^(-0.082 * 0.082 * t - C * 0.082)

Simplifying further, we get:

c(t) = 43.2 - e^(-0.082 * 0.082 * t - C * 0.082) / 0.082

where e is the base of the natural logarithm.