A medical technician is trying to determine what percentage of a patient's artery is blocked by plaque. To do this, she measures the blood pressure just before the region of blockage and finds that it is 1.20×104 Pa, while in the region of blockage it is 1.15×104 Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/s [.3 m/s], and the specific gravity of this patient's blood is 1.06.
What percentage of the cross-sectional area of the patient's artery is blocked by the plaque?
*I'm not even sure where to start for this problem. There are no examples like it in my book...*
To determine the percentage of the cross-sectional area of the patient's artery blocked by the plaque, we can use Bernoulli's equation for fluid flow. Bernoulli's equation relates the pressure, velocity, and height of a fluid flow.
The equation can be written as:
P + ½ρv² + ρgh = constant
Where:
P is the pressure of the fluid,
ρ is the density of the fluid,
v is the velocity of the fluid,
g is the acceleration due to gravity,
h is the height of the fluid.
In this case, we can ignore the height component since we are dealing with a horizontal flow. Therefore, the equation simplifies to:
P + ½ρv² = constant
Now, let's apply Bernoulli's equation to the situation at hand. The constant in this case will be the same before and after the blockage, as we are dealing with the same fluid. So, we can write:
P1 + ½ρv1² = P2 + ½ρv2²
Where:
P1 is the pressure just before the blockage,
P2 is the pressure in the region of the blockage,
v1 is the velocity just before the blockage,
v2 is the velocity in the region of the blockage.
We are given the pressures and the velocity just before the blockage. Rearranging the equation, we have:
P1 - P2 = ½ρ(v2² - v1²)
Now, let's solve for the ratio of the cross-sectional areas using the following relation:
A2/A1 = v1/v2
Where:
A1 is the cross-sectional area just before the blockage,
A2 is the cross-sectional area in the region of the blockage.
Rearranging the equation, we have:
A2 = (A1 × v1)/v2
Now, substituting the value of (v1/v2) from the previous equation, we get:
A2 = (A1 × (P1 - P2)/(2P1ρ))
Finally, the percentage of the cross-sectional area blocked can be calculated as:
Percentage Blocked = [(A1 - A2)/A1] × 100
Now that we have the equations, let's substitute the given values and calculate the percentage of the cross-sectional area blocked by the plaque.