The blood vascular system consists of blood vessels (arteries, arterioles, capillaries, and veins) that convey blood from the heart to the organs and back to the heart. This system should work so as to minimize the energy expended by the heart in pumping the blood. In particular, this energy is reduced when the resistance of the blood is lowered. One of Poiseuille's Laws gives the resistance R of the blood as

R=C(L/r^2)
where L is the length of the blood vessel, r is the radius, and C is a positive constant determined by the viscosity of the blood. (Poiseuille established this law experimentally, but it also follows from Equation 8.4.2.) The figure shows a main blood vessel with radius r branching at an angle into a smaller vessel with radius s. (Note in your answers use t instead of and r and s respectively instead of and .)

(a) Use Poiseuille's Law to find the total resistance of the blood along the path ABC where a and b are the distances shown in the figure. (Your answer may depend on a, b, C, r, s, and t.)
(c) Find the optimal branching angle when the radius of the smaller blood vessel is 3/3 the radius of the larger vessel. =

To find the total resistance of the blood along the path ABC, we can first calculate the resistances of the two branches separately and then combine them.

Let's denote the length of the larger blood vessel (AB) as L1 and the radius as r. The resistance of this vessel can be given as R1 = C(L1/r^2).

Similarly, let's denote the length of the smaller blood vessel (BC) as L2 and the radius as s. The resistance of this vessel can be given as R2 = C(L2/s^2).

The total resistance along the path ABC can be obtained by adding the resistances of these two vessels:

Total resistance (R_total) = R1 + R2 = C(L1/r^2) + C(L2/s^2).

Now, let's substitute the given information: the smaller blood vessel radius is 3/3 the radius of the larger vessel (s = (1/3)r).

Therefore, the total resistance can be written as:

R_total = C(L1/r^2) + C(L2/[(1/3)r]^2).

Simplifying further:

R_total = C(L1/r^2) + C(L2/(1/9)r^2).

R_total = C(L1/r^2) + 9C(L2/r^2).

R_total = C[L1/r^2 + 9L2/r^2].

R_total = C[(L1 + 9L2)/r^2].

So, the total resistance along the path ABC is C times the sum of lengths L1 and 9L2, divided by the square of the radius r.

To find the optimal branching angle, we need to consider the relationship between the radii of the two vessels.

Let's denote the branching angle as angle theta.

Based on the information given, the radius of the smaller vessel (s) is (1/3) times the radius of the larger vessel (r).

In terms of trigonometry, we can express this relationship as:

sin(theta) = (1/3).

To find the optimal theta, we need to solve this equation for theta.

sin(theta) = (1/3).

Taking the inverse sine (or arcsin) of both sides:

theta = arcsin(1/3).

Therefore, the optimal branching angle when the radius of the smaller blood vessel is 3/3 the radius of the larger vessel is arcsin(1/3).