Final answer:
We can analyze the given information, The average radii of the runner's blood vessels have increased by a factor of 2.0.
Step-by-step explanation:
In this scenario, we are asked to determine the factor by which the average radii of the runner's blood vessels have increased during a marathon race. To solve this, we need to consider the relationships between blood flow, blood viscosity, blood pressure, and vessel radius.
Using the equation for the flow rate of a fluid through a vessel, Q = (ΔP * π * r^4) / (8 * η * L), where Q is the flow rate, ΔP is the pressure difference, r is the vessel radius, η is the fluid viscosity, and L is the vessel length, we can analyze the given information.
Since the blood flow has increased to 10.0 times the resting rate, we can set up the equation: 10 = (ΔP * π * (r * F)^4) / (8 * η * L), where F is the factor we are trying to find. Similarly, the blood's viscosity dropping to 95.0% can be represented as η = 0.95 * η_0, where η_0 is the normal viscosity. Lastly, the blood pressure difference increasing by 50.0% can be represented as ΔP = 1.5 * ΔP_0, where ΔP_0 is the normal pressure difference.
Substituting these relationships into the flow rate equation, we get 10 = ((1.5 * ΔP_0) * π * (r * F)^4) / (8 * (0.95 * η_0) * L). Simplifying and rearranging the equation, we find F^4 = (10 * 8 * (0.95 * η_0) * L) / ((1.5 * ΔP_0) * π).
Calculating this expression, we get F = 2.0. Therefore, the average radii of the runner's blood vessels have increased by a factor of 2.0.