Homework SourseA drug with concentration 008gcc is deliverd into an organ a
A drug with concentration 0.08g/cc is deliverd into an organ at the rate of 12cc/sec and is dispersed at the same rate. Suppose the organ has volume 600cc and initially contains none of the drug.
(a) set up and solve a differential equation for the amount of drug A(t) in the organ at time t.
(b)Wt\'s the concentration of the drug in the organ after 30secs? After 1 min?
(c) How long does it take for the concentration of drug in the organ to reach 0.06g/cc?
(a) set up and solve a differential equation for the amount of drug A(t) in the organ at time t.
(b)Wt\'s the concentration of the drug in the organ after 30secs? After 1 min?
(c) How long does it take for the concentration of drug in the organ to reach 0.06g/cc?
Solution
rate of gms of drug coming inside the organ = (0.08 g/cc)*(12 cc/sec) = 0.96 g/sec fluids containing the drug are lost at rate of 12 cc/sec from 600 cc organ. now, let A(t) be gms of drug in the organ at any time t so, A(t) = drug initially present + drug which comes in - drug which flows out after time t has passed drug initially present = 0 drug that has come inside = (0.96t) gms drug that has flown out = [A(t)/600]*(12)*t so the balance yields A(t) = 0 + 0.96t - [A(t)*t]/50 or, A A(t) = 0.96t/(1 + 0.02t) so the differential equation is the derivative of A(t) A\'(t) = [0.96(1 + 0.02t) - (0.96t)*(0.0192t)]/(1 + 0.02t)^2 A\'(t) = [ -0.018432t^2 + 0.0192t + 0.96 ]/(1+0.02t)^2 b) after t = 30 secs A(t) = 18 gms so conc = A(t)/600 = 0.03 gm/cc after t = 60 secs A(t) = 26.2 gms so conc = 26.2/600 = 0.04364 gms/cc c) when conc = 0.06 gm/cc amount of drug in organ = (0.06*600) = 36gms = A(t) put in in equation of A(t) and solve for t you will get t = 178.6 sec or 2 min 59 secs
