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## Homework Statement

A certain drug is being administered intravenously to a hospital patient. Fluid containing

5 mg/cm

^{3}of the drug enters the patient’s bloodstream at a rate of 100 cm

^{3}/h. The drug

is absorbed by body tissues or otherwise leaves the bloodstream at a rate proportional to

the amount present, with a rate constant of 0.4 (h)

^{−1}.

(a) Assuming that the drug is always uniformly distributed throughout the bloodstream,

write a differential equation for the amount of the drug that is present in the bloodstream

at any time.

(b) How much of the drug is present in the bloodstream after a long time?

## Homework Equations

N/A

## The Attempt at a Solution

To find the rate of change, I know I have to use the amount in minus the amount out. The amount in seems to be 500t, and the amount out is 0.4/t (t is hours), but this gives me the formula [itex]y'=500t-\frac{0.4}{t}[/itex], which seems way too simple. The solved equation is [itex]y=250t^{2} - 0.4ln(t)[/itex]. This would also mean, for part b, that the patient has ∞ mg/cm

^{3}in her blood at time t=0. I'm sure I'm just being stupid and missing something simple here, but what am I doing wrong?

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