Blood flow A typical human heart pumps 20 mL of blood with...

Blood flow A typical human heart pumps 20 mL of blood with each stroke (stroke volume). Assuming a heart rate of 60 beats / min, a reasonable model for the outflow rate of the heart is $V^{\prime}(t)=20(1+\sin (2 \pi t)),$ where $V(t)$ is the amount of blood (in milliliters) pumped over the interval $[0, t], V(0)=0,$ and $t$ is measured in seconds. a. Graph the outflow rate function. b. Verify that the amount of blood pumped over a one-second interval is $20 \mathrm{mL}.$ c. Find the function that gives the total blood pumped between $t=0$ and a future time $t>0.$ d. What is the cardiac output over a period of 1 min? (Use calculus, then check your answer with algebra.)

Blood flow A typical human heart pumps 20 mL of blood with each stroke (stroke volume). Assuming a heart rate of
60 beats / min, a reasonable model for the outflow rate of the heart is $V^{\prime}(t)=20(1+\sin (2 \pi t)),$ where $V(t)$ is the amount of blood (in milliliters) pumped over the interval $[0, t], V(0)=0,$ and $t$ is measured in seconds.
a. Graph the outflow rate function.
b. Verify that the amount of blood pumped over a one-second interval is $20 \mathrm{mL}.$
c. Find the function that gives the total blood pumped between $t=0$ and a future time $t>0.$
d. What is the cardiac output over a period of 1 min? (Use calculus, then check your answer with algebra.)

Key Concepts

Rate Functions and Graphs

This concept involves understanding functions that describe the rate at which a quantity changes over time. Graphing such functions, especially those that exhibit periodic behavior like sine functions, helps in visualizing the fluctuations in the process and identifying key features such as peaks, troughs, and average values.

Definite Integration for Accumulated Change

Definite integration is a fundamental tool in calculus that allows us to calculate the total accumulation of a quantity from its rate of change over a specified interval. It is essential for modeling scenarios where it is necessary to determine the total output or change, such as the cumulative volume produced in a periodic process.

Periodic Functions and Sine Waves

Sine waves are a classic example of periodic functions characterized by their regular and repeating pattern. They are used to model phenomena that repeat at regular intervals, with key properties including amplitude, frequency, and phase shift. Understanding these properties is critical in translating real-world periodic processes into mathematical models.

Cardiac Output Measurement

Cardiac output is the measure of the total volume of blood pumped by the heart in a given time period. This concept combines the ideas of rate and accumulation, as it is typically determined by multiplying the stroke volume (the amount of blood pumped per beat) by the heart rate. It is a fundamental concept in physiology that uses mathematical modeling to assess cardiovascular health.

Transcript

00:01 Okay, here we're given a outflow rate of a heart, which is measured v prime of t equals 20 times 1 plus sign of 2 pi t.

00:13 And we're going to figure out a bunch of different things.

00:15 So the first thing i'd ask us to do is graph this.

00:17 So i'm going to go ahead and just use the stesnos calculator to graph that.

00:47 So we get a graph that looks something like that.

00:52 You can see that we have a lot of ups and downs.

00:56 Our blood flow rises very, blood flows at a very rapid and differentiating pace.

01:04 So the second thing that asks us to do is figure out how, what the blood flow is over one second, a one second interval.

01:12 So let's do from zero to one.

01:13 We have 20 times one plus sine of 2 pi t.

01:20 It's going to be equal to 20 times t minus 20 cosine.

01:30 Of 2 pi t divided by 2 pi from 0 to 20 or sorry from 0 to 1 so yeah so we're going to figure that up so 20 times 1 minus 20 cosine of 2 pi divided by 2 pi minus 20 times 0 plus or sorry minus 20 20 times 0 plus or sorry minus 20 20 times cosine of 2 pi times 0, which is just 0 over 2 pi.

02:14 If we add, if we subtract this, this becomes an addition symbol.

02:19 We have 20 cosine, negative 20 cosine 2 pi over 2 pi, 20 cosine 0.

02:25 Cosine of 0 is equal to 1.

02:26 Cosine of 2 pi is also equal to 1.

02:29 So these two cancel.

02:30 And we're left with just 20 milliliters.

02:32 The next question asks us, find the function.

02:35 We already did most of the work here.

02:37 So we know that our base function is going to be 20t minus 20 cosine of 2t over 2 pi plus c.

02:51 We're going to figure out what c is...

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