solve-the-given-problems-the-concentration-c-in-m-g-c-m3-of-a-certain-drug-in-a-patients-bloodstream

Question

Solve the given problems.
The concentration $C$ (in $m g / c m^{3}$ ) of a certain drug in a patient's bloodstream is $C=\frac{d .15 t}{t^{2}+1},$ where $t$ is the time (in $h$ ) after the drug is administered. Sketch the graph.

Aron Dotson · Accepted Answer

for problem 89. We&#x27;re looking at a rational functions. See, if t is equal to five t over T squared plus one or tea is greater than equal zero. And what it represents is the concentration of a drug over time after administration. Okay, so we can definitely go through. Our typical analysis of this function on the X intercept is when the actual function of zero. So it&#x27;s when the the numerator is zero. And that happens when t A zero. Okay, so you see, I already have the graph up here. The text. What best say to use graphic device. Um, so here it is. The orange reddish graph Here is the actual function I explained that blew ours outline here in a second BC. It does go through the origin of the Y, intercept his labs when you plug in zero. And so that would equal zero as well. So we have that on the craft. Vertical assam tote. Um, that is depends on the degree of the numerator and denominator, since a decree of the numerous less than the Curia Dominator that would imply that the horizontal acid is the ex axes like a zero Okay, so we have a horizontal ask himto, um, of the x axis, and you see that the rational function does approach that had an affinity. Um Well, sorry. It was like that is the horrors I&#x27;ll ask himto vertical assam tote. There is none. Okay, sack, for messing that up. That was the orders on last until the bird a glass and toast. There is none. Since you cannot make the denominator zero. Anytime you square tea, it&#x27;ll be positive. And one that&#x27;s positive. So there is no vertical. Ask himto here and you see it in the crap. Oh, Max can take on any value here. There is no vertical. Ask himto. Okay, so we&#x27;re asked a few things here. Definitely. Using the graphing utility makes this, um quite a bit easier. Um, in part, a were asked what is the maximum amount of concentration in the bloodstream so it occur at a maximum? Uh, and using a graphing utility like best most. You can just click on this point, or some graphing utility is going to find a function maximum and and use that to find it. But that would occur. Um, at the 0.1 comma 2.5, Which means after one year of time in this case, hours, uh, the maximum concentration that in the bloodstream be 2.5 and the units are milligrams per liter. Okay, Bart, be asked what happens to the amount of drug in the in the bloods Assist bloodstream over time. And so we&#x27;re looking at the red graph. What happens after many, many, many hours? We see that it approaches zero. Now, according to this model, we could never way. We would say it would never totally disappear because it does not approach. It doesn&#x27;t equal zero. It approaches, it gets infinitely small. Okay. And lastly, part see, um asked when would the amount of blood stream So when would see a t be less than 0.3 milligrams pre leader? Okay, so now this blue line comes And what I what I did here was I actually graft. Um, just half attacks equals 0.3. That&#x27;s this blue line here. Okay. And so what I What I was able to do then, is to just click somewhere on this craft and this dock came up right here, which is where the rational function crosses the blue eye, and that happens at 16.6 about grounded 0.3. And what that tells me is that after 16.6 hours approximately, it will be equal 2.3. So answer the question that will occur when the time is greater than 16.6 hours. Okay, after that, the red line will be below, and you start to see here in the graph a little bit below 0.3 milligrams per liter.

Jamie M · Answer

So we&#x27;re given this function to model concentration over time and were given these constraints A B and K or positive. Constance and B is greater than K. So I just picked these random numbers that do work for these constraints. And I plugged him into the function and they got this function. So if you want to grab that, the first thing I&#x27;m gonna look at is the first derivative. So see of tea, See, prime of tea. So the first derivative is gonna be negative e to the power of negative T plus to e to the negative to t. Okay. And I&#x27;m gonna set this equal to zero to find my critical points. And I see that c is equal to Ellen of two. All right, so Ellen of two is approximately 20.693 So if we test values, this is Ellen, too. If we test values, um, into our first derivative for the first derivative test for less than Ellen to, we&#x27;re going to see that it&#x27;s increasing and anything greater than is gonna be decreasing. So it&#x27;s a local Max. All right, So this could be useful for later and then see double prime toe. Look a ah kong cavity. See, double prime of tea. We&#x27;re going to get e to the negative. T minus four e to the negative to t If we said that equal to zero, we&#x27;re going to see, um, t equals to Ellen too, which is approximately, um, 1.39. And if we do the second derivative test of double prime and we test values around 1.39 we see that, um, it&#x27;s gonna be con cave down. And then Khan gave up. So this is an inflection point. All right, so we&#x27;re going to use this while we graph in just a moment, but I&#x27;m gonna look at ah, one more thing, because our graph actually does have an accent. Oh, it has. Ah. Horizontal ass in tow. So let&#x27;s test, um, the limit, as are variable t approaches infinity over function, which is one times e to the negative one t minus e to the negative to t. All right. And we&#x27;re gonna So let&#x27;s just say that we do, um, the limit as t approaches infinity, I&#x27;m gonna rewrite this in a way that looks easier to understand. So this is the same thing as one over e to the t because it&#x27;s, ah, negative exponents bring into the denominator. So one over E to the T minus one over each of the two tea. So this is the same thing, all right. And if you take a look at this, it&#x27;s every substitute. Wherever we see tease, we&#x27;re gonna substitute infinity. So as t approaches infinity, we see that one over. A very large number is close to zero. So this is gonna approach zero. So as T approaches infinity, our why values, which is concentration over time, is gonna approach zero. So we have a horizontal ask himto at zero. Okay, And now we can graph. And since we&#x27;re looking at time, I&#x27;m just gonna show the positive part of this cause negative time doesn&#x27;t really make sense to me. So this is concentration overtime for this access. And this is just time over here, and we know that we do have to say this is approximately one. We know we have a maximum of Ellen, too. And if you plug in Ellen to into the original function, you&#x27;ll see that it&#x27;s equal to 1/4. So we know that about Ellen two and 1/4 is our maximum and the sister of sketch. So I&#x27;m gonna say this is Ellen too. 1/4 at this point. And this is our maximum. Okay? And we see that, um, according to the second derivative test that it&#x27;s gonna be conquered down. And then Khan gave up in our inflection point is 1.39. So that&#x27;s what we&#x27;re gonna seethe. Sign change at about 1.39. So let&#x27;s say it&#x27;s right there. Okay? So it&#x27;s gonna be con cave down. Oh, and we also haven&#x27;t Don&#x27;t forget about the ass in tow. The horizontal asking to is right here. Okay, so we have, um, con cave down. It&#x27;s gonna first of all, increase until it hits are maximum local, maximum, and then it&#x27;s gonna starts a decrease after hits the max, but it&#x27;s still gonna be con cave down until it hits our inflection point where it&#x27;s gonna be Khan gave up an approach, but never touch are asking too. And this is our graph. All right, so this is our graph for concentration over time, and the question asked asks us, um, what&#x27;s happening in this graph. So we see as timing is increasing. So as time goes on, um, concentration increases until we hit the. So as time goes on, concentration increases until we hit the maximum concentration, and then after that, the concentration continues to decrease forever.

Ernest Castorena · Answer

get this time. They tell us that the concentration, uh, drugs in a registry is given by over is X squared minus two. Yeah. And of course, this is for time. So it will be for X greater than zero. So they inject you with some kind of drug, and then they monitor how the drug concentration, how heavy it is in your blood over time. Okay, so if we graft this, we can kind of see the behavior of his concentration, and I have already gone ahead and graft it on. What we can see is that the concentration peaks after a short time, and then it drops very subtly. Someone quickly on. That takes kind of, ah, long time, but it will eventually return to zero concentration. But there are measurable levels in it, or a decent amount depending on what kind of tools they have available. Measurable. Quite the Concord. Cool

Will Erickson · Answer

mother. Then this problem in part a were asked to graft the function f And so if we do that, um, can use a graphing calculator for this. I think you might as well we will see that the curve we did look, something like this. Some start to zero and its peak and then start tapering off. And as for units, if you look at the graph, the peak is a little less than 200. 200 is about here, and the peak occurs head about. It looks like 25. So, based on this than in part A, they ask, uh is F prime of 15 positive or negative prime of 15. While to get this list is look at the graph around 15. So 15 is somewhere around there. If we look right here, if we draw a little tangent line, Slope is clearly positive. That&#x27;s all we need to know. The graph is increasing at 15 therefore, F crime of 15 as positive. And then they ask us the same thing about 45. So let&#x27;s find 45. So are over here, and by the same logic, we draw little tangent line at 45. The slope is definitely negative. Or, in other words, the graph is decreasing at 45. So therefore, of prime of 45 it was negative. So that&#x27;s all for part A. For part V, we are asked to find f of 30 and then f prime of 30. So half of 30 Will those plug in 30 for tea into our function here. So 20 times 30 e to the minus 0.4 times 30 Huh? All right. In other words, 600 e has 0.4 times 30 and witnesses a calculator to get the decimal value here, which is 180.717 How they ask us to interpret this so the 30 was a T value and T is given in minutes. So all this means is that one tea equal 30 minutes. The drug concentration is 180.717 milligrams per milliliter. Just tells us the amount of the concentration for the derivative F prime of 30 will first need to take the derivative and then we can plug in 30. So let&#x27;s take the derivative of our function f so going back to the top. Let&#x27;s notice our function. F It&#x27;s 20 t is one function of f f t and we&#x27;re multiplying that by another function of tea. So we will need the product rule. Product rule tells us to take the derivative of first function multiplied by the second function and then plus the first function times the derivative of the second function. Okay, the derivative of 20 times t is just 20. That blue stuff here in just a plus 20 t can stay in the derivative of E to the minus 200.0 40. Well, we will need a chain rule here since it&#x27;s e to some function of teams of the derivative is the same thing as we started with E to the minus 0.0 40. But then we need to multiply by the derivative of this exponents, which is negative 0.4 Okay, so in other words, 20 e minus 0.0, for tea and over here within, clean things up a little bit. So we&#x27;ll have minus 20 times 0.4 You just think that&#x27;s the same thing as two times point for In other words, 0.0.8 t money. The minus point of afford t. So if we plug in 30 we&#x27;ll see the f prime of 30 equals. Well, let&#x27;s just put in 30 for all the cheese. Minus 0.8 times 30 times e to the negative Prideaux for times 30. And this will give us negative 1.205 And the units here, the units of CR milligrams per mo leader. But now this is the derivative. So we&#x27;re well, we&#x27;ve just found a rate. So let&#x27;s make sure we write this as milligrams from the leader her a minute because tea is given in minutes. All this means is that after 30 minutes, the concentration is changing at this rate. In other words, the concentration is, you can say decreasing at a rate of negative 1.205 milligrams. Fur militar permanence and the

Cinsy Krehbiel · Answer

So safety is a formula for the concentration of a drug in your system T minutes after it has been injected. Our job is to graph it and then see what the end behavior is or what happens as time goes on. So this is a rational function. The first thing we would do is reduce it. It can&#x27;t be reduced. There are no factors we can cancel out. So next we&#x27;re going to determine the Assam toads. Vertical assam toads happen when the denominator is equal to zero. I think we can see that there are no riel teas but will solve it just for proving. So subtract two from both sides. Take the square root of both sides. We get that T equals plus or minus the square root of negative too, which is imaginary. So there are no vertical Assen toes. Next thing is horizontal accent. Oh, so horizontal Assam tot&#x27;s tell us what happens as t gets increasingly large and we can see as t approaches infinity. This function is essentially gonna look like one over tea. All right, The the coefficients are not gonna be interesting in that little constant to is not gonna be very interesting. As T is super large, it&#x27;s one over T approaches. Zero. So we have a horizontal Assam tote at, um, why equals zero so we could go ahead and put that on there. There we go. That&#x27;s our horizontal acid tote. And what next we can we normally, we would look for the zeros and we would look for the y intercept, but it&#x27;s not meaningful for time to be negative. So the Y intercept is kind of boring. It&#x27;s just zero, right? So we know a T equals zero Y equals zero. So we have a point right there. Studios in blue, maybe. And it&#x27;s not going across the X axis again. That wouldn&#x27;t be meaningful for a concentration to go negative. So all that&#x27;s left for us to do is to make a little table of values so that we can make a graph four points and see what the shape is as it approaches the Assam toad. Now we know what the fire reaches, what it&#x27;s gonna be like. It&#x27;s gonna hug the Assam tone from the top. But typically, functions like this will do some interesting things at the beginning. So we&#x27;re gonna figure out a couple of values of tea and CFT. Basically some coordinate pairs and then we&#x27;ll try to take a stab of the shape. So when tia zero, there is nothing in there, right? And maybe right before the injection happens, then when t is one, we get 30/3. CFT is tense. We could plot that one 10. And how about when tea is, too? We&#x27;re going to get, um, but see, 60/2 squared is four plus two. It&#x27;s six. That&#x27;s another 10. So where Uh huh Now the chances that it dips back down, slim to none and the chances that it goes straight across, also slim to none. So we&#x27;re gonna go ahead now and say, Well, let&#x27;s see what happens at halfway between 1.5 so that we&#x27;re gonna use a calculator for and we get 10.6. So somewhere in between there and 10.6, etc. We have four. So it&#x27;s just a little past the half, right on our t. It&#x27;s gonna be above there and we don&#x27;t know for sure if if that&#x27;s right. Exactly. You know where the maximum is. But without calculus, we&#x27;re not gonna worry about that. Maximum. Alright, How about let&#x27;s try for we just see how quickly it&#x27;s approaching that Assam toad. So when we have four again, we&#x27;re gonna need a calculator than the concentration comes out to 6.7. So now we&#x27;re starting to drop down. So 6.7 is just shy of the third Dick Clark. And when it&#x27;s eight than the concentration is 3.6 again, thank you calculator. So here&#x27;s eight. 3.6 is just past the half. Yeah, we could start to see the shape materializing now and then. How about let&#x27;s try 15 and 20? So when Excess 15 calculator says to 15 and two and then exits 20 calculator says 1.5, and that&#x27;s pretty much now worth appointed diminishing returns. So we we get the just here, right? We got a 21.5. So now we&#x27;re gonna try to very carefully sketch in the shape, so it&#x27;s gonna come up fairly sharply. It&#x27;s going to be smooth. There&#x27;s a bump. Appear somewhere. Oops, we missed. We come down like this and it&#x27;s going to approach the ex. Access horizontally approaches this Assam Choate and somewhere out in the far reaches, the value will approach zero. So as time goes by right as T approaches infinity, the concentration approaches zero.

Problem

Solve the given problems. The combined capacitanc…

Problem 24 Easy Difficulty

Solve the given problems.
The concentration $C$ (in $m g / c m^{3}$ ) of a certain drug in a patient's bloodstream is $C=\frac{d .15 t}{t^{2}+1},$ where $t$ is the time (in $h$ ) after the drug is administered. Sketch the graph.

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. EvansBasic Technical Mathematics with Calculus

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Allyn J. Washington, Richard S. Evans

Basic Technical Mathematics with Calculus