1. A patient in hospital is on an IV drip. This delivers a saline solution with

Question

1. A patient in hospital is on an IV drip. This delivers a saline solution with an added antibiotic. The mixture is delivered at a rate of 125ml per hour (that is 3 l per day). The concentration of the antibiotic is 31.25mg per 125 ml (that is 250 mg per l). The patient has 7l of blood, and the body's organs remove liquid at the same rate it is added by the drip. The liquid removed contains the antibiotic in the same concentration as in the blood. a. Find the formula for the concentration of antibiotic in the blood at a time t hours after the start of the drip. b. At what time does the concentration of antibiotic in the blood reach 100mg per l? Please show all the work! Thank you!!!

Explanation / Answer

Take y(t) = the amount of antibiotic in the patient's blood at time t (mg) ; Since the flow in (qi) and the flow out (qo) are equal, the volume is constant V(t) = V and the concentration of antibiotic in the blood at time t is y(t)/V ; V = 7 l ; qi = qo = 3 l/day ; the concentration of the solution entering ci = 250 mg/l ; the concentration of the solution eliminated co = y/V (the same concentration as in the blood) ; The net rate of change of the amount of antibiotic in the blood is dy/dt ; The equation is dy/dt = qi * ci - qo * co Since 3 l/day = 1/8 l /hour dy/dt = 1/8 * 250 - 1/8 * y/V dy/dt = 250/8 - y/56 You need to solve the equation assuming y(0) = 0 ; y(t) = 1750 (1 - e^(-t/56) ) , when t is hours ; The concentration would be then c = y/V : c(t) = 250 (1 - e^(-t/56) )

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