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Problem 20 Medium Difficulty

High blood pressure results from constriction of the arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseuille's Law to show that if $ R_0 $ and $ P_0 $ are normal values of the radius and pressure in an artery and the constricted values are $ R $ and $ P $, then for the flux to remain constant, $ P $ and $ R $ are related by the equation
$$ \frac{P}{P_0} = (\frac{R_0}{R})^4 $$
Deduce that if the radius of an artery is reduced to three-fourths of its former value, then the pressure is more than tripled.

Answer

$\approx 3.2 P_{0}$

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Video Transcript

so the problem access to use was a slaw to determine a relation between normal values and constricted values for Flo, um, in an artery. So since the flow rate is to remain constant between the two, we can plug the normal and constricted values into pauses law and set the results equal. That will give us that pie. P zero are zero to the fourth, and that would be equal to P R to the fourth over the same denominator. So here we can multiply through by the entire denominator and by pie, since in both of them on both sides, they're the same. And so the entire denominator cancel. And of course, these pies will cancel. That will give us that. P zero are zero to the fourth is equal to P R to the fourth. And so if we divide by p zero and divide by art of the fourth, well, yeah, that p divided by p zero is equal to are zero. Is it am I? And since both of those air to the fourth, we can put it on the outside like that. And so now we want to know what happens when the construction is to 3/4 of the original radius. So in this case, we get that are constricted radius this three over four or a zero. So in this case, using this expression that we just arrived weaken, find that p over P zero. We equal to our zero over 3/4 are zero onto the fourth. So these two are zeros will actually cancel, and three and one divided by three or four is the same. This for over three. So that will be for over three to the fore, which is 2 56 over 81 or approximately 3.2. So now, if we multiply by p zero, we get that P is approximately 3.2 p zero. So that means that yes, the conclusions are correct that when you have 3/4 the size of the original radius as your constricted radius pressure more than triples. So here is that result. And then here is the derived expression