Problem 23 Easy Difficulty

The graph of the concentration function $ c(t) $ is shown after a 7-mg injection of dye into a heart. Use Simpson's Rule to estimate the cardiac output.

Answer

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Video Transcript

in this question, we're going to go over cardiac output given a concentration function that is presented as a graph. And we are not given a nice function for this, but we will find a way we can usually approximate the area under a graph. Using Simpson's rule the area under the graph, of course, is an integral. Let's write out what our integral will be now from the graph that's in the question. We know that it's going to be an integral from 0 to 16 and it will be the integral from 0 to 16 of the concentration function is approximately going to be will have a change in time over three being multiplied by F of zero F at T0 Plus four times f of T one Plus two times f of T two and so on, and so forth. So we'll have four times F of TN -1 minus f of the last time. That's our formula now. Yeah, Well, we'll estimate with 16 terms. So that means that we're going to have a long integral to watch. But because we have 16 terms and we're integrating from 0-16, we can use we can take our change in time to just be one second, So we'll have 1/3. Okay. Multiplied by f of zero plus four times F of one Plus two times f of two and so on. Up until we have four times f 15 minus this would be please or both places, Sorry? Plus F of six teams. Uh huh. Right. But what are all of these values from the graph? It looks like we will be having one third. Yeah. Being multiplied by. Okay. F of zero. Well F of zero is zero F of one looks to be approximately three. Approximately four F of two appears to be six F of three appears to be seven Fo four appears to be about 7.3 F of five appears to be 7.1 FF six appears to be approximately 6.8 f of seven appears to be six F of eight appears to be 5.3 FF nine appears to be 4.8 F of 10 appears to be four. FF- 11 appears to be 3.5 F of 12 appears to be three. FF- 13 appears to be 2.5 F of 14 appears to be 2.1 After 15, appears to be 1.9, And FF 16 appears to be just 1.5. And putting this in your calculator should return approximately 72 0.57 and this is in milligrams, times seconds per liter. Okay, now that we have our concentration total, we're now set to find the cardiac output, so We know that f is going to be equal to, we know that the initial injection was seven mg. So we have a seven up top and we'll just divide this by 72.57. And the units will properly cancel out because we know that this is in milligrams. This is in milligrams seconds for leader. Sure. And this will result. And of course About 0.0964, leaders per second. which will multiply by 60 seconds to get rid of Which is multiplied by 60 seconds per minute. And this will result in about 5.78 leader per minute. Let me make that seven better. And that is the end of this question.