When blood flows along a blood vessel, the flux F (the volume of blood per unit time that flows past a given point) is proportional to the fourth power of the radius R of the blood vessel:
F = kR4
(This is known as Poiseuille's Law.) A partially clogged artery can be expanded by an operation called angioplasty, in which a balloon-tipped catheter is in
ated inside the artery in order to wide it and restore the normal blood flow.
Show that the relative change in F is about four times the relative change in R. How will a 1% increase in the radius a�ffect the flow of blood?
A 1% increase in the radius corresponds to a/an ______% ______ in blood flow.
I got the answer. It is 4 % increases in blood flow.
thanks anyway.
To find the relative change in F (relative change in blood flow) compared to the relative change in R (relative change in the radius), we can differentiate the equation F = kR^4 with respect to R.
Taking the derivative of F with respect to R, we get:
dF/dR = 4kR^3
To find the relative change in F (dF/F), we divide the change in F (dF) by the original value of F:
(dF/F) = (dF)/(kR^4)
Next, we can find the relative change in R (dR/R), which is the same as the percentage change in R:
(dR/R) = 1% = 0.01
Now, we can substitute these values into the equation:
(dF/F) = (dF)/(kR^4) = (4kR^3 * (dR/R))/(kR^4)
Canceling out k and R^3, we have:
(dF/F) = 4 * (dR/R)
This shows that the relative change in F is approximately four times the relative change in R.
Now, let's calculate how a 1% increase in the radius affects the flow of blood:
(dF/F) = 4 * (dR/R)
(dF/F) = 4 * 0.01 = 0.04
Therefore, a 1% increase in the radius corresponds to a 0.04 or 4% increase in blood flow.
To show that the relative change in F is about four times the relative change in R, we can start by considering the equation F = kR^4, where F represents the blood flow and R represents the radius of the blood vessel.
Let's assume we have an initial radius R and corresponding blood flow F. Now, let's increase the radius by a small amount, ΔR, resulting in a new radius R + ΔR. We can calculate the new blood flow F' using the equation F' = k(R + ΔR)^4.
To determine the relative change in F, we can find the ratio of the change in F to the initial value of F:
Relative change in F = (F' - F) / F
Substituting the values of F and F', we get:
Relative change in F = [k(R + ΔR)^4 - kR^4] / (kR^4)
Simplifying this expression, we have:
Relative change in F = [(R + ΔR)^4 - R^4] / R^4
Using the binomial expansion formula, we can expand (R + ΔR)^4:
(R + ΔR)^4 = R^4 + 4R^3ΔR + 6R^2(ΔR)^2 + 4R(ΔR)^3 + (ΔR)^4
Substituting this expansion into the relative change in F expression, we get:
Relative change in F = [(R^4 + 4R^3ΔR + 6R^2(ΔR)^2 + 4R(ΔR)^3 + (ΔR)^4) - R^4] / R^4
Simplifying further:
Relative change in F = [4R^3ΔR + 6R^2(ΔR)^2 + 4R(ΔR)^3 + (ΔR)^4] / R^4
Now, if we assume that ΔR is small, then terms involving higher powers of ΔR become negligible. Thus, we can ignore terms (ΔR)^2, (ΔR)^3, and (ΔR)^4:
Relative change in F ≈ 4R^3ΔR / R^4
Now, simplifying this expression, we have:
Relative change in F ≈ 4ΔR / R
Comparing this result to the original equation, we can see that the relative change in F (ΔF/F) is approximately equal to 4 times the relative change in R (ΔR/R):
ΔF/F ≈ 4(ΔR/R)
This confirms that the relative change in F is about four times the relative change in R.
To determine how a 1% increase in the radius affects the flow of blood, we can substitute the values into the derived expression:
ΔF/F ≈ 4(ΔR/R)
Here, ΔR/R corresponds to a 1% increase, which can be expressed as 0.01 (since 1% = 0.01).
Therefore, we have:
ΔF/F ≈ 4(0.01)
Simplifying this equation, we find:
ΔF/F ≈ 0.04
So, a 1% increase in the radius will result in approximately a 4% increase in blood flow.