Question

Poiseuille's Law states that the volume, V, of blood flowing through an artery in a unit of time at a fixed pressure is directly proportional to the fourth power of the radius of the artery. That is, V=kr4, where r is the radius of the artery y about how much must the radius of the artery be increased to achieve a 55 % increase in the volume of blood flowing through the artery?

Answer 1

Answer:

11.58%

Step-by-step explanation:

The initial volume if blood flowing through the artery is given by

$V=kr^4$

To achieve a new volume of 155% (55% increase) of the initial volume, the new radius must be:

$V'= 1.55V\\1.55V=k(r')^4\\1.55kr^4 = k(r')^4\\(\sqrt[4]{1.55}*r)^4=(r')^4 \\(1.1158*r)^4=(r')^4 \\r'=1.1158*r$

Since the new radius is 1.1158 times larger than the initial radius, the percentage increased is:

$I=(1.1158 - 1)*100\%\\I= 11.58\%$

Answer 2

Using proportions, it is found that the radius must be increased by 11.58%.

The equation for the volume is:

$V = kr^4$

To increase the volume, since the constant does not change, the radius has to be multiplied by a constant a, thus:

$V_i = k(ar)^4 = ka^4r^4$

We want an increase of 55%, thus:

$V_i = 1.55V$

$ka^4r^4 = 1.55kr^4$

$a^4 = 1.55$

$a = \sqrt[4]{1.55}$

$a = (1.55)^{\frac{1}{4}}$

$a = 1.1158$

1.1158 - 1 = 0.1158

0.1158 x 100% = 11.58%

The radius has to be increased by 11.58%.

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