Question

When the body requires an increased blood flow rate in a particular organ or muscle, it can accomplish this by increasing the diameter of arterioles in that area. This is referred to as vasodilation. What percentage increase in the diameter of an arteriole is required to double the volume flow rate of blood, all other factors remaining the same?
This is whats given in the answer box, I need a percentage.

d2-d1/d1=________%

Answer 1

Answer: The percentage increase in diameter is 19%

Explanation:

Using Poiseuille's law which states that Flow rate , Q = ΔPπr⁴/8ηl

where ΔP is pressure difference, π is a constant, r is the radius which is half of the diameter, η is viscosity of blood, l is length of blood vessel

Let Q₁ be the intitial flow rate, Q₂ the final flow rate,r₁ and r₂ the initial and final radius.

Note: r = d/2, therefore, r₁⁴ = d₁⁴/16 r₂⁴ = d₂⁴/16

Also, Q₂ = 2Q₁, since the flow rate is doubled

Since all factors except diameter is constant, d₂⁴/16 = 2d₁⁴/16

d₂⁴/16 =  d₁⁴/8

multiply both sides of the equation by 16

d₂⁴ = 2d₁⁴

taking the fourth root of both sides

d₂ = 1.19*d₁

d₂ - d₁ = 1.19d₁ - d₁

d₂ - d₁ = d₁(1.19 - 1)

d₂ - d₁ = 0.19d₁

d₂ - d₁/d₁ = (0.19d₁/d₁)*100%

(d₂ - d₁/d₁)*100% = 19%

Therefore, the percentage increase in diameter is 19%