(I) Calculate the average speed of blood flow in the major art...

Question

(I) Calculate the average speed of blood flow in the major arteries of the body, which have a total cross-sectional area of about 2.0 cm² . Use the data of Example 13–14.

Accepted Answer

Hi, everyone. Let's take a look at this practice problem dealed with fluid flow. In this problem, we need to determine the average speed of water flowing through a network of pipes with a combined cross sectional area of 3.5 centimeters squared. The main supply pipe has a radius of 1.5 centimeters and the water speed is 30 centimeters per second. In the main pipe, we're given four choices here. Choice A is 0.4 m per second. Choice B is 0.61 m per second. C is 4.9 m per second and choice D is 7.4 m per second. Now, since we're dealing with fluid flow, let's recall our continuity equation. The continuity equation says that for a fluid moving through a pipe with a cross sectional area, A, if I multiply that area A by the speed of the fluid B, then this has to equal a constant. So I have a V is equal to a constant. I don't know what the value of that constant is. However, but it is a constant. That means if I pick any two points in this fluid flow through these pipes, then their area multiplied by their speed have to be equal to each other. So that's what we're going to do here. We're gonna pick two points. It's common points, one and two. And so I'll have a, one multiplied by V one is equal to a two multiplied by B two. And here, um, 0.1 is going to be a point in this network of pipes whose speed I'm looking for And 0.2 will be a location in the supply pipe. So here and at this point, I need to identify what information I was given. So first thing I need to identify is the area of my supply uh sorry of my network pipes. And I was given that in the problem and that is the 3.5 centimeters squared. So I'll go ahead and plug that in. So I have 3.5 centimeters squared multiplied by B one which is the speed I'm looking for. And that has to equal the cross sectional area of my supply pipe. However, the only thing I'm told is that I have a radius of 1.5 centimeters. So recall um the area of a circle since I have a radius, I have a circular cross section. So the area of the circle is just equal to pi R squared A equals pi R squared. So I'm gonna plug that in for a two. So I'll have I'll have the 3.5 centimeters squared multiplied by V one equal to pi. And now I'm going to multiply this by the radius is the 1.5 centimeters and don't want to square that radius. And then I'm left with the speed and of the water and the supply pipe. And that is given the problem and that is 30 centimeters per second, this 30 centimeters per second. Now, at this point, I need to solve for V one but notice that I've not converted any of my units into SI units. So I have left things in terms of centimeters squared and centimeters instead of converting everything to meters. And that's because a lot of times in these problems, since I'm gonna be doing a lot of division, uh the units will cancel out. So it's a good idea to wait until the very end, your final answer to do any conversions to SI units because a lot of the units will cancel out along the way. And so you're just saving some time by not doing the conversion till the end. Let's stop this. For V one, we have V one equal to, I have I multiplied by the 1.5 centimeters that's gonna get squared. And I will divide that by a 3.5 centimeters squared and that fraction is multiplied by the 30 centimeters per second. And if I look at the fraction, I see that I have centimeter centimeters squared on the denominator and centimeters squared in the numerator, those cancel out. And what I'm left with is just basically numbers. And so I can solve four V one. I wanna plug those values into my calculator. I end up with 61 centimeters per second. Now, at this point, I need to convert to SI units. And so I'm going to multiply my C 61 centimeters per second by the conversion factor of 1 m divided by 100 centimeters. That's because there's 100 centimeters in 1 m. And so that would give me V one in terms of meters per second. So how V one is equal to 0.61 meters per second? So that's gonna be my final answer and that corresponds to answer B so just a quick little recap. So in this problem, we use the continuity equation, you set up um this equivalence between the area and the speed for two different points. And one of the tricks that we looked at was not converting units until the very end. So that was able to save us a little bit of time. So I hope that this has been useful for you and I'll see you in the next video.

(I) Calculate the average speed of blood flow in the major arteries of the body, which have a total cross-sectional area of about 2.0 cm² . Use the data of Example 13–14.

Key Concepts

Average Speed

Cross-Sectional Area

Flow Rate

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