Question

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 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. 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 user instead of and r and s respectively instead of rand r2.) branchine

(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 1.) R = Cila-bcott)/(r^2)+(b
(b) Find the value of cose which minimizes the resistance. (Your answer may depend on a, b, C.r. and s.) cos 0 =
(c) Find the optimal branching angle when the radius of the smaller blood vessel is 1/1 the radius of the larger vessel. 0 = 63.61

Answer 1

Poiseuille's Law is used to analyze blood flow and vascular resistance in relation to the circulatory system, with variables like vessel radius and branching angles affecting efficiency and energy expenditure by the heart.

Step-by-step explanation:

The study of blood flow within the circulatory system can be analyzed using Poiseuille's Law, which relates vascular resistance to the length and radius of a blood vessel as well as the viscosity of the blood.

In the situation described, we have a branching system where the total resistance along the path ABC is being calculated, and the law can be applied to determine the effects of various factors, such as the vessel radius and the angle of branching, on this resistance.

The branching of vessels in the circulatory system is crucial for efficiently delivering blood to various parts of the body and this system demonstrates a remarkable ability to regulate flow by adjusting vessel diameters, all of which ties into Poiseuille's principles.

To determine the optimal branching angle and consequent resistance, it's necessary to combine mathematical principles with understanding of physiological concepts—particularly regarding how vessel diameter and length affect resistance and flow through the system.

The optimization of these variables ensures that the energy expended by the heart is minimized, maintaining an efficient circulation of blood throughout the body.