Problem 29 - Precalculus Mathematics for Calculus

Question

Blood Pressure 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 \(\mathrm{mmHg}\) 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 \(p ?\)

Step-by-Step Solution

Answer

(a) Amplitude=25, Period=1/80 min, Frequency=80 cycles/min. (c) Exercise increases frequency, decreases period.

Step 1: Identify the Amplitude

The amplitude of a sinusoidal function of the form \( p(t) = A + B \sin(Ct) \) is given by the absolute value of the coefficient of the sine function. Here, the amplitude is \( |25| = 25 \).

Step 2: Determine the Period

The period of a sine function \( \sin(Ct) \) is given by \( \frac{2\pi}{C} \). Therefore, for \( p(t) = 115 + 25 \sin(160\pi t) \), \( C = 160\pi \). So, the period is \( \frac{2\pi}{160\pi} = \frac{1}{80} \) minutes.

Step 3: Calculate the Frequency

The frequency is the reciprocal of the period. So, the frequency is \( \frac{1}{\text{Period}} = 80 \) cycles per minute.

Step 4: Sketch the Graph

To sketch the function \( p(t) = 115 + 25 \sin(160\pi t) \), note that the graph oscillates between \( 115 - 25 = 90 \) and \( 115 + 25 = 140 \) with a period of \( \frac{1}{80} \) minutes. Plot these key characteristics and draw the sinusoidal wave.

Step 5: Effect of Exercise on Period and Frequency

If a person exercises, the heart beats faster. This results in a higher frequency of the blood pressure cycle, which means the period will decrease. The general behavior indicates a more frequent oscillation during exercise.

Key Concepts

Amplitude

The amplitude of a trigonometric function, like sine and cosine, refers to the distance from the middle of the wave to its peak, or its trough. In simpler terms, we can think of amplitude as the height of the wave. In the function given in the exercise, \( p(t) = 115 + 25 \sin(160\pi t) \), the amplitude is represented by the coefficient of the sine term, which is 25.

• The sine term oscillates between -1 and 1.
• Therefore, multiplying it by 25 means the oscillation will be between -25 and 25.
• This oscillation is added to the base value of 115 mmHg, resulting in a pressure fluctuation between 90 mmHg and 140 mmHg.

Understanding the amplitude is crucial as it helps visualize and sketch the wave of the blood pressure function, providing insights into how much the blood pressure varies with each heartbeat.

Period

The period of a trigonometric function is the duration it takes for the function to complete a full cycle. In the context of the blood pressure function \( p(t) = 115 + 25 \sin(160\pi t) \), the period can be found using the formula for the period of a sine function, \( \frac{2\pi}{C} \), where \( C \) is the coefficient of \( t \) in the sin function.

Calculating the Period

To calculate the period of \( p(t) \):

• Identify \( C = 160\pi \).
• The period is \( \frac{2\pi}{160\pi} = \frac{1}{80} \) minutes.

This means that one complete cycle of blood pressure oscillations takes \( \frac{1}{80} \) minutes, which is a very rapid process, reflecting the continuous and rhythmic heartbeat. The short period indicates the rapidity of each heartbeat, showing how quickly the blood pressure rises and falls.

Frequency

In trigonometric functions, frequency is the number of complete cycles a function goes through in a unit of time. It is essentially the inverse of the period and indicates how often the waves repeat. When it comes to the function \( p(t) = 115 + 25 \sin(160\pi t) \), understanding frequency tells us about the speed of oscillations per minute.

Understanding Frequency

• The frequency is calculated as the reciprocal of the period.
• Given a period \( \frac{1}{80} \) minutes, the frequency becomes \( 80 \) cycles per minute.

This high frequency means that the wave pattern, or oscillations of blood pressure, occurs 80 times each minute. During exercise, this frequency increases, indicating that the heart is beating faster. Thus, the concept of frequency is essential in analyzing how quickly events, like a heartbeat in this case, occur over a period of time.