Final answer:
The dilation of blood vessels during a marathon results from physiological adjustments to increase blood flow as described by Poiseuille's Law. By setting up an equation based on this law that reflects the increased blood flow, decreased blood viscosity, and increased pressure difference, one can calculate the factor by which the blood vessel radii have increased.
Step-by-step explanation:
During vigorous physical activity, such as a marathon race, the human body makes several physiological adjustments to meet the increased demand for nutrients and oxygen in the muscles. One such adjustment is the dilation of blood vessels to increase blood flow. The relationship between blood flow (Q), the blood vessel radius (r), blood viscosity (η), and the pressure difference (ΔP) can be described by Poiseuille's Law, which states that Q is directly proportional to r^4ΔP/η.
In this scenario, the blood flow increases by 15.0 times, the blood viscosity decreases to 90.0% of its resting value, and the pressure difference increases by 50.0%. To find out by what factor the radii of blood vessels have increased, we set up the equation reflecting these changes based on the relations described by Poiseuille's Law: (Q2/Q1) = ((r2/r1)^4) × (ΔP2/ΔP1) × (η1/η2), where '1' and '2' denote initial and changed conditions, respectively. We can then solve for (r2/r1), the factor by which the average radii have increased.
We start by isolating (r2/r1):
(r2/r1)^4 = (Q2/Q1) × (η2/η1) / (ΔP2/ΔP1)
Substituting the given values:
(r2/r1)^4 = (15.0) × (0.9) / (1.5)
Solving for (r2/r1), we find that the average radii of the blood vessels have increased by a certain factor that can be determined through algebraic manipulation.