Part 1 of 4 A patient receives an injection of a drug. After...

Part 1 of 4 A patient receives an injection of a drug. After the injection, the amount of drug in the bloodstream is halved every 40 minutes. Suppose that immediately after the injection, the amount in the patient's bloodstream is 16 mg/liter. After 40 minutes, how much will be in the patient's bloodstream? mg/liter.

Transcript

00:01 Now in this question it is given that a drug is being injected to the patient's body and the concentration of the drug is 200 milligrams per liter and the rate of injection is 40 milliliter per hour.

00:19 Now the volume of blood remains constant which is 4 liter in the patient's body.

00:25 Now the drug is metabolized by the patient at the rate 10 % per hour.

00:32 Now what we have to find, we have to find that how much drug is found in the patient's body after 75 minutes.

00:41 And we have to also assume that there is no drug in the system before receiving the treatment.

00:49 So this question is based on the differential equation.

00:54 So let's see how to solve this question.

00:57 Now first we're going to find that the rate of increase of drug.

01:11 In the patient's body now it will be equals to now the consideration of drug is 200 milligram per liter and the rate of injection is 40 milliliter per hour so that would be equals to 200 times this lead we can do one thing we can convert the unit this is a milligram per liter times this would be 40 times 10 to the power minus 3 liter per hour now here this one will cancel so it will be equals to 8 milligram per hour so this is the increase of a drug in the patient's body now next i'm going to find the rate of decrease of drug in the patient's body that would be equals to now the the drug is metabolized at the rate 10 % per hour so that would be 10 % means 0 .1 times a now what is x here x is the concentration of the drug in the system at any time t so this is x now next we can write that the rate of change of concentration of drug that is d x over dd would be equals to increase of drug minus the decrease so that would be 8 minus 0 .1 now point 1 we can write this thing as 1 over 10 times x now we can rewrite this differential equation as dx over dt plus 1 over 10 x is equals to 8 now if you see carefully this differential equation is of the form first order linear different linear differential equation and the standard equation we have dy over dx plus py is equals to q and the solution of this differential equation is given by y times the integrating factor is equals to integration q times the integrating factor is equal to integration q times the integrating factor plus c now what is integrating factor integrating factor here is integration a to the power sorry e to the power integration p d x this is the way we find we find the integrating factor so let's use this formula here to solve our differential equation so this would be now let's first find the integrating factor that would be e to the power integration now what is p here p is 1 over 10 so 1 over 10 and dx would be replaced with dt so that would be equals to a to the power 1 over 10 t now the solution would be y times no here this y should be replaced with x so x times integrating factor is a to the power 1 over 10 t is equals to integration now q we have 8 so 8 times integrating factor is e to the power 1 over 10 t then dot d t plus c now we have to evaluate this integration to solve the question...

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