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Chapter 5: Problem 41

Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. 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 ) at time \(t,\) measured in minutes. (a) Find the amplitude, period, and frequency of \(p\) (b) Sketch a graph of \(p\) (c) If a person is exercising, his or her heart beats faster. How does this affect the period and frequency of \(p ?\)

Short Answer

Expert verified
(a) Amplitude: 25, Period: \(\frac{1}{80}\) min, Frequency: 80/min. (b) Graph: Sine wave between 90 and 140 mmHg. (c) Increased heart rate decreases period and increases frequency.

Step by step solution

01

Identify the Amplitude

The amplitude of the function is the coefficient of the sine function, which in this case is given by the number 25. This amplitude indicates the maximum deviation from the average value (115 mmHg) of the blood pressure.
02

Determine the Period

The period of a sine function is calculated using the formula \( \frac{2\pi}{b} \), where \( b \) is the coefficient of \( t \) inside the sine function. Here, \( b = 160\pi \), so the period is \( \frac{2\pi}{160\pi} = \frac{1}{80} \) minutes. This is the time it takes for one complete cycle of the pressure variation.
03

Calculate the Frequency

Frequency is the reciprocal of the period. Since the period is \( \frac{1}{80} \) minutes, the frequency is \( 80 \) cycles per minute, meaning the heart beats 80 times a minute under this condition.
04

Sketch the Graph of p(t)

To sketch the graph, plot the sine wave with an amplitude of 25 and a period of \( \frac{1}{80} \) minutes centered around 115 mmHg. The graph oscillates between \( 115 - 25 = 90 \) mmHg and \( 115 + 25 = 140 \) mmHg.
05

Exercise Effect on Period and Frequency

When a person exercises, the heart rate increases. This means the function \( p(t) \) would have a larger coefficient for \( t \), decreasing the period and increasing the frequency. Therefore, the heart beats more often, resulting in more cycles per minute.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Amplitude
Amplitude is an essential concept when analyzing trigonometric functions like sine and cosine. In our blood pressure model, the amplitude is the number 25, which is the coefficient in front of the sine function. This value represents the maximum distance the pressure can vary from its mean value, which is 115 mmHg in this problem.
  • The greater the amplitude, the more significant the variation around the mean.
  • For example, an amplitude of 25 means that the blood pressure varies by up to 25 mmHg above or below the average.
This means that the person's blood pressure can peak at 140 mmHg and drop to 90 mmHg with each heartbeat. Understanding amplitude is critical in interpreting how intense the fluctuations in a cyclic process like the heartbeat are.
Period
The period of a trigonometric function tells us the duration of one complete cycle of the wave. For sine and cosine functions, the period is calculated as \(\frac{2\pi}{b}\), where \(b\) is the coefficient of the variable \(t\) inside the sine function.
  • In our example, \(b = 160\pi\), leading to a period of \(\frac{2\pi}{160\pi} = \frac{1}{80}\) minutes.
This result indicates that it takes \(\frac{1}{80}\) of a minute for the blood pressure to complete one full cycle of increase and decrease.
The concept of period is crucial for understanding how long it takes for repetitive events, such as heartbeats, to complete a cycle. With a shorter period, the cycle occurs more frequently.
Frequency
Frequency is closely related to the period. It refers to how many complete cycles occur in a unit of time, typically expressed in cycles per second (or minute in this case).
  • It is the reciprocal of the period.
  • In the example provided, the frequency is \(80\) cycles per minute.
This means the heart is beating 80 times within a minute. When considering exercise or other conditions that alter the heartbeat rate, frequency gives a direct measure of how often the cycles (heartbeats) are happening.
An increased frequency, such as during exercise, means the heart beats more often within a given time.
Graphing Trigonometric Functions
Graphing trigonometric functions like \(p(t) = 115 + 25 \sin(160\pi t)\) provides a visual understanding of how these functions behave over time. When graphing:
  • Start by identifying the amplitude, period, and mean value.
  • Plot the baseline at the mean value, which is 115 mmHg in this case.
  • The sine wave will oscillate around this mean within the amplitude range from 90 mmHg to 140 mmHg (115 卤 25).
  • One complete cycle of the waveform occurs over the period length, which is \(\frac{1}{80}\) minutes for this scenario.
Graphing helps visualize the regularity and variability of the cyclic events, which is especially helpful in physiological measurements like blood pressure over time.
Modelling with Trigonometric Functions
Trigonometric functions can be used to model cyclical phenomena in science and engineering, like blood pressure variations. These functions are suitable because:
  • They naturally oscillate, making them ideal for representing periodic events.
  • The parameters of amplitude, period, and frequency can be adjusted to fit specific models, demonstrating versatility.
  • By changing coefficients, they can model increased rates, such as during exercise when heartbeats become more frequent and regular.
Using a trigonometric model like our example, we can gain insights into how factors such as exercise impact physiological metrics like blood pressure. This approach allows scientists and engineers to predict behaviors, understand patterns, and analyze cycles efficiently.

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