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Problem 2 Medium Difficulty

A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after $ t $ minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.
$$
\begin{array}{|l|c|c|c|c|c|}
\hline t \text { (min) } & 36 & 38 & 40 & 42 & 44 \\
\hline \text { Heartbeats } & 2530 & 2661 & 2806 & 2948 & 3080 \\
\hline
\end{array}
$$
The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient's heart rate after 42 minutes using the secant line between the points with the given values of $ t $.
(a) $ t = 36 $ and $ t = 42 $
(b) $ t = 38 $ and $ t = 42 $
(c) $ t = 40 $ and $ t = 42 $
(d) $ t = 42 $ and $ t = 44 $
What are your conclusions?

Answer

A.$$69.6 \frac{\text {heartbeats}}{\min }$$
B.71.75$\frac{\text { heartbeats }}{\text { min }}$
C.71$\frac{\text {heartbeats}}{\min }$
D.66$\frac{\text {heartbeats}}{\min }$

Discussion

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Video Transcript

for Problem number two and Stuart eighth Edition, Section two point one. It says a cardiac monitor Eyes used to measure their heart rate of a patient after surgery. It compiles the number of heartbeats after teen minutes when the data on the table are graft. The slope of the tangent line represents the heart rate and beats per minute. The monitor estimates this value the slow or the hard way by calculating the slope of a picket line. S o. Use the data given to estimate the patient's heart rate after forty two minutes, using the secret line between the points with the given values of tea. So let's pull up again. The table from this problem showing you at every time is shown a thirty six minutes thirty minutes forty, forty two and forty for. It gives you the total amount of heartbeats that were measured. And you're going to use this data as long along with this slope formula to calculate the heart rate. So you take the difference in the heartbeat between two sets of points and they divided by the difference in the time Onda. Again, the problem describes how this method of finding the soap is how you calculate the heart rate and beats permanence on. We noticed that imparts A, B, C and D. The reference is always the same. It's t equals forty two. So this is the main reference for calculating the heart rate. And we I can pull up the spreadsheet and the spreadsheet has the data from this table here, and it allows us to see to figure out first the difference in the heartbeat between each of you, these points and the reference time. Forty two minutes. Because that's the a time of interest. We see the difference in heartbeats. We divided by the difference in time that's calculated from this table, and this last column gives us the slopes. So for part A between the time miss, thirty six minutes in time is forty two minutes. We see that there's a difference of four hundred eighteen heartbeats. Between those two totals. We know that forty two is six minutes after thirty six. That's the difference in the time we divide those two numbers, and we get the answer for Party, which is sixty nine point six, repeating that her pits permanent. We do the same for thirty eight minutes, giving us seventy one point seventy five beats per minute for party attending goes forty minutes to get, the seventy one will be permitted to value as the answer at forty minutes in part. See, finally, party is between t equals forty two minutes and tickles forty four minutes. We see that the slope or the beats per minutes calculated, is sixty six beats per minute. On that. Those are the answers to Ports A, B, C and D. The final question asks us where our conclusions, because the point of the purpose of this problem is, too, has to meet the patient's heart rate after forty forty two minutes. So this is the no key goal, and one method is to take the slope calculated before forty two minutes, the number closest to forty two before which is at forty. And then we also take the slope calculated after forty two minutes, just right after at forty four minutes, and we titty carriage of these two numbers. If we do that, we should be able to get an estimate for the beats per minute at forty two minutes. So anyone sixty six is about two hundred and thirty seven decried it. Right, too, gets us a estimate of sixty point five beats permanent, and that is our conclusion that that is the heart rate at forty two minutes.