Relating portions of Poiseille's Law to concepts of pressure/radius

[oddjobmj]

Homework Statement

High blood pressure results from constriction of arteries. To maintain a normal flow rate (flux), the heart has to pump harder, thus increasing the blood pressure. Use Poiseille's Law to show that if R₀ and P₀ are normal value sof 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

P/P₀ = (R₀/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.

Homework Equations

P/P₀ = (R₀/R)^4

and what I know of Poiseille's Law

F = ((pi)(P)(R^4))/((8)(n)(l))

Where R=volume, P=pressure, l=length of blood vessel, F=flux, and n=viscosity of blood

The Attempt at a Solution

Firstly, I've worded and formatted the question exactly as listed. I did this because I'm a little confused at what it's even asking me to do. Does it read like there are two parts; 1) Proving P and R are related by the given equation and 2) deducing the given specific relationship between a change in radius and a change in volume?

I really don't know where to start. I don't want the answer given to me but any hints on how to get started and what they're really asking me for would be greatly appreciated.

 

[lanedance]

so the initial radius R₀ and pressure P₀ give a flux F
$$F = \frac{\pi P_0 R_0^4}{8nl}$$

we assume at a new reduced radius R, we can find a pressure P which gives the same flux F
$$F = \frac{\pi P R^4}{8nl}$$

now equate the RHS of each and rearrange to get P/P₀

 

[oddjobmj]

Thank you for your reply Lanedance. The way you worded it makes much more sense. However, I'm not sure what you mean by RHS.

Two instances of the equivilant of F will result in all known variables to cancel out on both sides leaving the variables we're working with. Now I see how they derive the first equation I listed. Now if I plug in some reasonable values I can solve for the unknown.

I guess I'm mostly confused because this now seems like a rather simple algebra problem and this is a calculus II class.

EDIT: Also, would you recommend I place reasonable values for R and P in and solve or should I leave it as is with all the variables?

 

[lanedance]

yeah it looks like algebra mainly to me

RHS - right hand side

personally i always prefer to leave the variables in until the last step or as late as possible. I find it more flexible & more audiatble

the question asks you to subsititute in $R = \frac{3}{4}R_0$

 

[oddjobmj]

Awesome, after inserting the 3/4 and adjusting all to the RHS I essentially came down to 256/81=P/4000 (4000=P₀, the value I picked as a common P value; used in a previous example).

Then I simply said 256/81 > 3 and did a little explaining.

Thank you for your help!