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 1.20×10^4 Pa, while in the region of blockage it is 1.15×10^4 Pa. Furthermore, she knows that blood flowing through the normal artery just before the point of blockage is traveling at 30.0 cm/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 know that I have to convert the velocity to 0.300 m/s, then specific gravity is equal to the density of the blood in relation to the density of water, so the density of blood is 1060 kg/m^3. I have no idea where to go from here. I would try putting it into the equation:
P1 + (density * gravity * y1) + (.5 * density * (velocity1)^2) = P2 + (density * gravity * y2) + (.5 * density * (velocity2)^2)
But that doesn't seem to work. How do you figure out what the blockage amount is? I don't want an answer, I just want to know how.
y1=y2, so the equation goes to
P1+.5density*V1^2 =P2+.5density*V2^2
solve for v2.
then, use the law of mass continuity
Area1*velocity1=area2*velocity2
and solve for area2/area1, which will be a decimal number (then convert it to percent). That subtracted from 100 will the blocked area.
To determine the percentage of the cross-sectional area of the patient's artery that is blocked by the plaque, you can use the Bernoulli's equation for fluid flow. Here's how you can proceed step-by-step:
Step 1: Convert the velocity units
Since the velocity is given as 30.0 cm/s, you need to convert it to m/s. There are 100 cm in 1 m, so the velocity would be 0.3 m/s.
Step 2: Calculate the pressure difference
The pressure just before the region of blockage (P1) is given as 1.20×10^4 Pa, and in the region of blockage (P2) it is 1.15×10^4 Pa. The pressure difference (ΔP) can be calculated as (P1 - P2).
Step 3: Calculate the altitude difference
To calculate the altitude difference (Δy), you need to know the specific gravity of the blood (ρb) and the acceleration due to gravity (g). In this case, the specific gravity of the blood is given as 1.06. The altitude difference can be calculated as (ρb * g * Δy).
Step 4: Plug the values into Bernoulli's equation
Bernoulli's equation can be written as:
P1 + (ρb * g * y1) + (0.5 * ρb * v1^2) = P2 + (ρb * g * y2) + (0.5 * ρb * v2^2)
In this case, P1 is the pressure before the blockage, P2 is the pressure in the blockage region, y1 is the altitude before the blockage, y2 is the altitude in the blockage region, v1 is the velocity before the blockage, and v2 is the velocity in the blockage region.
Step 5: Rearrange the equation
Rearrange the equation to isolate the altitude difference:
Δy = (P2 - P1) / (ρb * g) - (0.5 * (v2^2 - v1^2) / (g))
Step 6: Calculate the blockage percentage
The percentage of the cross-sectional area of the artery blocked by the plaque can be calculated using the formula:
Blockage percentage = (Δy / y1) * 100%
Remember to substitute the values you have calculated or been given into the formula to determine the blockage percentage.
Note: Make sure to check and convert all units to maintain consistency throughout the calculations.
To determine the percentage of the artery that is blocked by plaque, we need to apply Bernoulli's equation, which relates the pressure, velocity, and height at two different points in a fluid flow system. The equation can be written as:
P1 + 0.5ρv1^2 + ρgh1 = P2 + 0.5ρv2^2 + ρgh2
where P1 and P2 are the pressures at points 1 and 2, ρ is the density of the fluid (in this case, blood), v1 and v2 are the velocities at points 1 and 2, g is the acceleration due to gravity, and h1 and h2 are the heights at points 1 and 2.
In this case, point 1 is just before the region of blockage, and point 2 is inside the region of blockage.
Let's break down the equation and solve step by step:
1. Convert the velocity from cm/s to m/s:
Given: v1 = 30.0 cm/s
To convert cm/s to m/s, divide by 100: v1 = 30.0 / 100 = 0.300 m/s
2. Calculate the difference in pressure:
Given: P1 = 1.20×10^4 Pa, P2 = 1.15×10^4 Pa
ΔP = P1 - P2 = (1.20×10^4) - (1.15×10^4) = 0.05×10^4 Pa = 0.05×10^4 N/m^2 = 0.05×10^4 J/m^3
3. Calculate the specific gravity:
Given: Specific gravity = 1.06
Specific gravity = ρ_blood / ρ_water
ρ_blood = Specific gravity * ρ_water
Given: ρ_water = 1000 kg/m^3
ρ_blood = 1.06 * 1000 = 1060 kg/m^3
4. Substitute the values into Bernoulli's equation:
0.05×10^4 = 0.5 * 1060 * (0.300)^2 + 1060 * 9.81 * h2 - 1060 * 9.81 * h1
5. Rearrange the equation to solve for the height difference (h2 - h1):
h2 - h1 = (0.05×10^4 - 0.5 * 1060 * (0.300)^2) / (1060 * 9.81)
6. Calculate the percentage of the cross-sectional area blocked:
Percentage blocked = (h2 - h1) / h2 * 100
By following these steps and performing the calculations, you can determine the percentage of the patient's artery blocked by plaque.