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 \dfrac{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. The figure shows a
main blood vessel with radius r branching at an angle
\theta into a smaller vessel with radius s . (Note in your
answers use t instead of \theta and r and s respectively
instead of r_1 and r_2 .)

(a) Use Poiseuille's Law to find the total resistance of the
blood along the path

(b) Find the value of

(c) Find the optimal branching angle when the radius of the
smaller blood vessel is

You can earn partial credit on this problem.