Final answer:
The factor by which a blood clot has reduced the radius of an artery can be calculated using Poiseuille's law, which relates flow rate, pressure difference, and radius. By solving the equation using the given values of flow rate and pressure difference increase, we find that the clot has reduced the radius by a factor of approximately 0.55.
Step-by-step explanation:
To determine the factor by which a blood clot has reduced the radius of an artery, we need to consider the relationship between flow rate, pressure difference, and radius. According to Poiseuille's law, flow rate is inversely proportional to the radius of the artery raised to the fourth power. Additionally, flow rate is directly proportional to the pressure difference.
In this case, the flow rate has been reduced to 10.0% of its normal value, which means the radius of the artery must have reduced by a certain factor. If we assume the pressure difference has increased by a factor of 1.2 as given in the question, we can calculate the factor by which the radius has reduced.
Using the relationship between flow rate and radius, we have:
(new_flow_rate)/(original_flow_rate) = (new_radius)⁴/(original_radius)⁴
Given that the new flow rate is 0.1 times the original flow rate and the pressure difference has increased by 20.0%, we can substitute the values into the equation:
0.1 = (new_radius)⁴/(original_radius)⁴
0.1 = (new_radius/original_radius)⁴
Taking the fourth root of both sides, we get:
∛0.1 = new_radius/original_radius
new_radius = ∛0.1 x original_radius
new_radius ≈ 0.548 x original_radius
Therefore, the factor by which the blood clot has reduced the radius of the artery is approximately 0.548 or can be rounded to 0.55.