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Problem 43 Hard Difficulty

A glucose solution is administered intravenously into the bloodstream at a constant rate $ r. $ As the glucose is added, it is converted into other substances and removed from the bloodstream at a rate that is proportional to the concentration at that time. Thus a model for the concentration $ C = C(t) $ of the glucose solution in the bloodstream is
$ \frac {dC}{dt} = r - kC $
where $ k $ is a positive constant.
(a) Suppose that the concentration at time $ t = 0 $ is $ C_o. $ Determine the concentration at any time $ t $ by solving the differential equation.
(b) Assuming that $ C_o < r/k, $ find lim $ _{t \to \infty} C(t) $ and interpret your answer.

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Related Courses

Calculus 2 / BC

Calculus: Early Transcendentals

Chapter 9

Differential Equations

Section 3

Separable Equations

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A PHYS1034 students is asked to model the cockroach population found in his/her dormitory. The student assume that the population of cockroach is growing at a constant rate proportional to the initial population. The student wants to account for the carry

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Video Transcript

So let's start with problem number eight. Given a glucose administrated intervene ministry into the bloodstream at a constant rate Art and concentration concentration off blue Coach. Well, at I m d. That is fine. This place the given different decision. Think CBT devils too? Our miners came to see where he either All the tea Constant Now we have to find the content is not the blue coat at time t So we have to solve this different selling goods, right? So what is this off a minus is he is people to people and also it is given that also do human that at the university Geo he off video is a video at the concentration of the group of that time you know is CEO that is given. So now we have to follow the difference allegation And I have to find the concentration at 19 Internet in both sides we get what is that the CEO to see BC by R minus K c is the 1st 20 p This is t t. This implies that t is he goes toe minus one by key minus one like d C into our mine skc right so whatever I do have, um, multiplying by minus one bikey and sk the positive constant we can It is perfectly relevant. No, this is the technique. What is physically see your pussy. Okay. P reversal minus one by K. And it is log mode R minus one by Casey Did the in Julia tonight? Because it is d of our lineup. How is in the limit? We get a seed, you don't proceed. And then this is my Skippy. This is love of more ar minus. Is he, uh, minus love off our miners video. So from this, we get it to the par minus Katie, because ar minus K. C by r minus case he not That's fine. So are miners case not into the bar. Linus, Katie is off on a minus K c On this, we get, uh, yeah, so K C. Because toe ar minus auto minus g is it will be for nine s. Katie, We found this. They get see? Because you are like a class. He geo minus r k. We bar minus Katie. Dividing this by key is a cesium minus our wiki. So this is the concept of the next time he and our answer your physical to our cases. You don't mind our kids you don't minus cases. This gives the equation of the concentration of the loop of that time T Now come to the question. Never be this particular toe. Find the limit on limit details to infinite C or P Degree are Mikey. So that goes to this. Goes to the u R C. You lived in a very positive constant video. Uh, there's an R V i k and so 30 depart the oneness kitty goes to zero as limited as you, you industry impunity going on 80 in and here we can, uh oh, me. Lt's modular thing, as are by key that Did you get out and see you. So here we can omit this model is as this is positive. So and this is a concentration. So here we can omit the model results. So that's why we have here on this model s as this is given. Okay?

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Problem

A certain small country has 10 billion dollars in…