The resistance of blood flow in a blood vessel (R)

Question

The resistance of blood flow in a blood vessel (R) is inversely proportional to the fourth power of the radius (r) of the respective blood vessel

a. Build a function based on the given information.

b. What would be the domain of the function you created? (Remember to think in terms of this question). Give answer in interval notation

c. In terms of this question, describe in words what the dependent variable of the function you created is

d. In terms of this question, describe in words what the independent variable of the function you created is

e. Based on the function you created, describe why it is or isn't possible to have a blood vessel that has zero (0) resistances of blood flow

f. Graph the function you created

Answer 1

a) When two things are inversely proportional, it means that one value decreases at the same rate that the other increases. In this problem, it means that as our blood vessel radius increases, the resistance of flow decreases. However, a distinction must be made. The problem states that the resistance is inversely proportional to the fourth power of the radius, not just the radius by itself.

The common way to write a general function for inverse proportionality would be y = k/x, where x is the input or independent variable, y is the output or dependent variable, and k is some defined constant of proportionality. However, as stated above, the two things that are inversely proportional are not resistance and radius, but resistance and radius raised to the fourth power. So this general function can be modified as follows:

Let y = resistance (R)

Let x = radius to the fourth power (r⁴)

R = k/r⁴, where k = constant of proportionality

b) To answer this part, we just need to think logically about the physical limitations to our question. Can we have a blood vessel radius that is less than zero? Likewise, can we have a blood vessel radius that is infinitely large? In both cases, the answer is no. To be more specific, the size of our blood vessels will have some realistic upper bound (nobody has a 1ft radius blood vessel). It will also have some realistic lower bound. Really, the interval should be the range between this lower and upper bound. You may have to do some research to quantify what this is if you want to use actual numbers in your answer.

c) Based on the above explanation, the dependent variable would be the resistance, R.

d) Based on the above explanation, the independent variable would be the blood vessel radius, r.

e) Let's set our resistance to zero, R = 0.

0 = k/r⁴

To make this equation true, r would have to be infinity. It does not logically make sense to have a blood vessel with an infinite radius.

f) The exact shape of the graph will depend on the proportionality constant, k. But you can get a general idea by graphing y = 1/x⁴.