Blood Pressure Each time your heart beats, your blood...

Blood Pressure Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120$/ 80$ is considered normal. A certain person's blood pressure is modeled by the function $$p(t)=115+25 \sin (160 \pi t)$$ where $p(t)$ is the pressure in mmHg (millimeters of mercury), at time $t$ measured in minutes.(a) Find the period of $p$ . (b) Find the number of heartbeats per minute. (c) Graph the function $p$ . (d) Find the blood pressure reading. How does this compare to normal blood pressure?

Blood Pressure Each time your heart beats, your blood pressure first increases and then decreases as the heart rests between beats. The maximum and minimum blood pressures
are called the systolic and diastolic pressures, respectively. Your blood pressure reading is written as systolic/diastolic. A reading of 120$/ 80$ is considered normal.
A certain person's blood pressure is modeled by the function
$$p(t)=115+25 \sin (160 \pi t)$$
where $p(t)$ is the pressure in mmHg (millimeters of mercury), at time $t$ measured in minutes.(a) Find the period of $p$ .
(b) Find the number of heartbeats per minute.
(c) Graph the function $p$ .
(d) Find the blood pressure reading. How does this
compare to normal blood pressure?

Key Concepts

Sinusoidal Functions

Sinusoidal functions, such as the sine function, are used to model periodic phenomena because they naturally repeat over regular intervals. They capture behaviors where a quantity oscillates above and below a central value, which is why they are ideal for simulating processes like the rhythmic beating of a heart or variations in blood pressure.

Amplitude

The amplitude of a sinusoidal function is the distance from the midline of the function to its maximum or minimum value. It represents the extent of variation in the oscillation. In the context of modeling, the amplitude shows how much the variable, such as blood pressure, deviates from its average value.

Vertical Shift

The vertical shift in a sinusoidal function is a constant added to the function that moves it up or down on the graph. This shift determines the midline around which the oscillations occur. In applications like blood pressure modeling, it represents the average or baseline level of the quantity being measured.

Period

The period of a sinusoidal function is the length of the interval required for the function to complete one full cycle of its pattern. It is determined by the frequency factor inside the sine function. Understanding the period is crucial in linking the mathematical model to real-world time intervals, such as the duration of one heartbeat cycle.

Frequency

Frequency is the number of cycles that occur in a unit of time. It is the inverse of the period and directly relates to how often a repeating event, such as a heartbeat, occurs. In mathematical models of periodic phenomena, frequency quantitatively describes the speed of the oscillation.

Graphing Trigonometric Functions

Graphing trigonometric functions involves plotting points based on key characteristics like amplitude, period, vertical shift, and phase shift. This process helps visualize the behavior of the function over time and is fundamental in analyzing periodic data, making it easier to interpret how the modeled quantity changes.

Blood Pressure Components

Blood pressure is typically measured using two values: systolic pressure, which is the maximum pressure during heart contraction, and diastolic pressure, which is the minimum pressure when the heart is at rest. Understanding these components is essential when modeling blood pressure, as they provide a clinical context to the oscillations described by the mathematical function.

Transcript

00:01 Okay, so for this problem, we're given a function that models a person's blood pressure over time, where t is the time of minutes and p of t is the pressure and milliliters of mercury.

00:10 For part a, we're as to find the period of this function p right here.

00:15 Now, if you were looking on the trick function's graph, you can see that the function is cyclical.

00:21 And a period by definition would just be the length of one cycle.

00:24 So it would be the length of this line right here.

00:29 Since this is how often the graph repeats itself.

00:33 Keep in mind, this is the graph of the cosine function, but for a sine function, it would just be the same thing where you look at the y -axis, you see where the graph begins, and you see how often until it repeats itself.

00:45 So for a sign function, the period would be the length of this line.

00:50 However, for this problem, we're not given a graph, we're just given this function right here.

00:55 So for that, you would first make sure that the function is written in state.

01:00 Standard form, which it is.

01:02 For the period, we're going to be looking at this b value right here, which we can see is 160 pi.

01:11 To get the period of any sign or cosine function, the formula that you need to know is 2 pi over b.

01:20 Plugging b in, that is 2 pi over 160 pi, which simplifies to 1 over 80.

01:32 And that is our period right here.

01:35 Again, this is the formula that you need to know, and just know that if the value of b is negative, you would take the absolute value.

01:43 Now, for part b, we're asked to find the number of heartbeats per minute.

01:48 This problem is basically asking us for the frequency, since the frequency is how often something occurs per unit of time.

01:57 For this, you need to know that frequency is the inverse of period.

02:02 Since the period in part a is 1 over 80, the frequency would be 80 beats per minute.

02:09 And that would be your answer for part b.

02:15 Now for part c, we are asked to graph the function.

02:19 For that, we're going to be looking at the standard form of a trick function again, and we're going to assign values to a, b, c, and d.

02:27 A right here is the amplitude, and we can see.

02:31 That that is 25...

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