The volume flow rate in an artery supplying the brain is 3.10 10-6 m3/s.
(a) If the radius of the artery is 4.50 10-3 m, determine the average blood speed.
1 m/s
(b) Find the average blood speed at a constriction in the artery if the constriction reduces the radius by a factor of 4. Assume that the volume flow rate is the same as that in part (a).
To determine the average blood speed, we can use the equation for the volume flow rate in an artery:
Q = A * v
Where Q is the volume flow rate, A is the cross-sectional area of the artery, and v is the average blood speed.
(a) Given:
Q = 3.10 * 10^(-6) m^3/s
r = 4.50 * 10^(-3) m
The cross-sectional area of the artery can be determined using the formula:
A = π * r^2
A = π * (4.50 * 10^(-3) m)^2
A ≈ 6.37 * 10^(-5) m^2
Now, we can rearrange the formula for the volume flow rate to solve for the average blood speed:
v = Q / A
v = (3.10 * 10^(-6) m^3/s) / (6.37 * 10^(-5) m^2)
v ≈ 0.0487 m/s
Therefore, the average blood speed is approximately 0.0487 m/s.
(b) Given:
Q = 3.10 * 10^(-6) m^3/s
r_constriction = r / 4
The cross-sectional area of the constriction can be determined using the formula:
A_constriction = π * (r_constriction)^2
A_constriction = π * [(4.50 * 10^(-3) m) / 4]^2
A_constriction ≈ 1.27 * 10^(-5) m^2
Now, we can use the formula for the volume flow rate to solve for the average blood speed at the constriction:
v_constriction = Q / A_constriction
v_constriction = (3.10 * 10^(-6) m^3/s) / (1.27 * 10^(-5) m^2)
v_constriction ≈ 0.244 m/s
Therefore, the average blood speed at the constriction is approximately 0.244 m/s.
To solve this problem, we can make use of the equation for volume flow rate in a pipe:
Q = A * v
where Q is the volume flow rate, A is the cross-sectional area of the pipe, and v is the average fluid velocity.
(a) To find the average blood speed in the given artery, we can rearrange the equation as follows:
v = Q / A
Given Q = 3.10 * 10^-6 m^3/s and the radius of the artery, we can calculate the cross-sectional area of the artery using the formula for the area of a circle:
A = π * r^2
where r is the radius of the artery. Plugging in the values, we get:
A = π * (4.50 * 10^-3 m)^2
Calculating A, we can substitute the values into the equation to find the average blood speed:
v = (3.10 * 10^-6 m^3/s) / A
Now we can solve for v:
v ≈ 1 m/s
Therefore, the average blood speed in the artery is approximately 1 m/s.
(b) Now let's consider the constriction in the artery, which reduces the radius by a factor of 4. This means the new radius of the constriction will be (4.50 * 10^-3 m) / 4.
Using the same equation as before, we calculate the new cross-sectional area of the constriction:
A' = π * [(4.50 * 10^-3 m) / 4]^2
To maintain the same volume flow rate from part (a), we know that Q = Q'. Therefore, we can rearrange the equation from part (a) to solve for v':
v' = Q / A'
Substituting the values, we have:
v' = (3.10 * 10^-6 m^3/s) / A'
Calculating A', we can solve for v' to find the average blood speed at the constriction:
v' ≈ 4 m/s
Therefore, the average blood speed at the constriction in the artery is approximately 4 m/s.
Velocity = Q/A, where Q is the volume flow rate.
If r is reduced by a factor of 4, area is reduced by a factor of 16, and velocity increases by that factor