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Problem 70 Hard Difficulty

Dialysis treatment removes urea and other waste products from a patient's blood by diverting some of the bloodflow externally through a machine called a dialyzer. The rate at which urea is removed from the blood (in $\mathrm{mg} / \mathrm{min}$ ) is often well described by the equation
$$
u(t)=\frac{r}{V} C_{0} e^{-r t / V}
$$
where $r$ is the rate of flow of blood through the dialyzer (in $\mathrm{mL} / \mathrm{min}$), $V$ is the volume of the patient's blood (in $\mathrm{mL}$ ), and $C_{0}$ is the amount of urea in the blood (in $\mathrm{mg}$ ) at time $t=0$. Evaluate the integral $\int_{0}^{\infty} u(t)$ and interpret it.

Answer

$C_{0},$ all urea would be removed if $t \rightarrow \infty$

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Video Transcript

Hi there. So for this problem we have some kind of bashing U. Of T. When she is following our older B is zero. The exponential of minus error ties the over the volume. So we know that art is the rate of law of glass through the dialyzers B is the volume and center to is the amount of urea in the blood at time T equals to zero. Dad when we need to calculate first is the integral from zero to positive infinity of this function. So we're going to do that. You're the infinity or the function UFT which is simply the discussion here and now we need to determine what of this. What are the of these quantities? Depends on the time Banks County. And we can see that the only turn to depends on T. Is the exponential function which has in its argument depends on T. So we can just we're all the rest outside the integral in which is simply need to integrate this afforded. We are integrated this in the body old Bt which is the only valuable in here D. T. To make easier the ism integral. We can't that that w we can we can do what we call a change memorials and then call w minus R. T over B. So that if we degree initiate this w we will obtain minus are over because that is a constant. The of the differential heat. No, we are going to put this information home so we can see from here. Uh and deal teams is my news the over our nine don't live you. So putting that we can see that that factor that will came from these differential goes away where they want outside injured role. Um they create this way we can see that this goes with this and this with this we all just off this and we will have the exponential of value. This integral of course is not defining these intervals. We could do that but which is simply going to solve the interval and later on um return to to the national volleyball team. So we know that um the integral of the exponential is just the exponential function. So it's that easy. So we will obtain the U. S. Financial on W. And we need to know who is W. So w is this in here? So we just put that and we need to evaluate in the initial Interpol's which is zero to positive infinity. So we're going to do that uh w is minus hard times the over B. And this is from zero to cool ass. Um infinity from zero. We know that the exponential function is just want so you know there's we need to, you know when you have all played an integral, you first emulated in the upper limit which is infinity. But when you put black community in here you can see that it is now the financial, it is the exponential of minus ability. And when you see the graph of the exponential function which is this when we go to minus infinity, Biggs function tends to be zero. So yes, that's zero and that is minus. We know that if the revelation in the upper limit minus the evolution in the and lower in it, so we'll take minus Heat of zero because that's the the lower limit and this science council and we will attain C. O. Um the o the exponential of zero. And you know the the exponential zero is want because in here it is one and will obtain the constant. These is zero. So where we are to obtain it is that the interval of this function give us the amount of urea in the blood at time zero. So that's they're not seeming amount of syria you can structure for block. So that is the interpretation of this office in the room, mounds of yuri, um our time, The Evil zero. So that's the amount of period, the initial period. So we have extracted all of that amount of furia when we enhance the time to infinity because that's the maximum we can strive from block. Thanks