Problem 77 - Algebra and Trigonometry

Question

Blood Pressure Each time your heart beats, your bloo 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, 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?

Step-by-Step Solution

Answer

Period of \( p(t) \) is \( \frac{1}{80} \) minutes, 80 heartbeats per minute. Blood pressure is \( 140/90 \), higher than normal \( 120/80 \).

Step 1: Understanding the Function

The function given is \( p(t) = 115 + 25 \sin(160\pi t) \). This is a sinusoidal function representing blood pressure over time in minutes. The baseline blood pressure is 115 mmHg, and it oscillates with an amplitude of 25 mmHg.

Step 2: Finding the Period

The function \( p(t) = 115 + 25 \sin(160\pi t) \) is in the form \( A + B\sin(Ct) \), where \( C = 160\pi \). The period \( T \) of a sine function \( \sin(\omega t) \) is given by \( T = \frac{2\pi}{\omega} = \frac{2\pi}{160\pi} = \frac{1}{80} \). So, the period of \( p(t) \) is \( \frac{1}{80} \) minutes.

Step 3: Finding Heartbeats per Minute

Since \( \frac{1}{80} \) minutes is the period of each heartbeat cycle, the number of cycles, or heartbeats per minute, is \( \frac{1}{\frac{1}{80}} = 80 \) heartbeats per minute.

Step 4: Graphing the Function

Graph \( p(t) = 115 + 25 \sin(160\pi t) \) using a graphing calculator or software. The graph oscillates between \( 115 + 25 = 140 \) mmHg and \( 115 - 25 = 90 \) mmHg, showing the blood pressure throughout the time.

Step 5: Finding Blood Pressure Reading

The maximum pressure of the function happens when \( \sin(160\pi t) = 1 \), which gives \( 115 + 25 = 140 \). The minimum happens when \( \sin(160\pi t) = -1 \), giving \( 115 - 25 = 90 \). So the blood pressure reading is \( 140/90 \).

Step 6: Comparing with Normal Blood Pressure

The calculated blood pressure reading \( 140/90 \) is compared to the normal \( 120/80 \). Thus, this blood pressure is higher than normal levels.

Key Concepts

Sinusoidal Functions

Sinusoidal functions, such as the one used in the blood pressure model, are mathematical functions that describe a smooth, repetitive oscillation. They are based on the sine or cosine functions, which are fundamental in trigonometry. These functions are used to model periodic phenomena, meaning they repeat at regular intervals. Examples include sound waves, light waves, and the alternating current in electricity.

The general form of a sinusoidal function is given by:

• \( y = A + B \sin(Ct) \)

- **\(A\)** represents the vertical shift or baseline level of the function.
- **\(B\)** is the amplitude, which indicates the peak deviation from the baseline.
- **\(C\)** determines the frequency of oscillation, thereby influencing the period.

• This element captures how fast the function completes one cycle.

In the context of blood pressure modeling, the sinusoidal function describes how blood pressure changes over time, rising and falling with each heartbeat, creating a wave-like pattern.

Blood Pressure Modeling

Blood pressure modeling employs sinusoidal functions to represent the cyclical nature of blood pressure changes as the heart beats. Each cycle of the sinusoidal wave corresponds to a heartbeat, with the peaks and troughs representing the systolic and diastolic pressures, respectively.

The specific function used in our example is:

• \( p(t) = 115 + 25 \sin(160\pi t) \)

- **Baseline level, 115 mmHg**: This is the average value around which the blood pressure oscillates.
- **Amplitude, 25 mmHg**: Indicates the difference between the average blood pressure and the extreme values (either high or low).

By analyzing the given sinusoidal function, we comprehend how the blood pressure deviates from its baseline with each heart contraction and relaxation. The formula helps predict blood pressure readings over time, providing insight into this essential cardiovascular metric.

Heartbeats per Minute

Understanding heartbeats per minute involves interpreting the period of the oscillation represented by the sinusoidal function. The period is the time it takes to complete one full cycle of the heartbeat. After deriving the period from the equation, you can calculate the heartbeats per minute.

For the function:

• \( p(t) = 115 + 25 \sin(160\pi t) \)

The frequency element \(160\pi\) helps us determine the period:

• The period \( T \) is calculated as \( \frac{2\pi}{160\pi} = \frac{1}{80} \) minutes.

To find the number of heartbeats per minute, take the reciprocal of the period:

• \( \frac{1}{\frac{1}{80}} = 80 \) heartbeats per minute.

This means the heart beats 80 times within one minute, which is typical for a resting adult.

Amplitude and Period

Amplitude and period are key characteristics of sinusoidal functions that define the shape and timing of the wave.

- **Amplitude**: It represents the height of the wave and is measured from the baseline to the peak, which corresponds to half the distance between the wave's high and low points. In blood pressure modeling, the amplitude (25 mmHg) indicates the extent of blood pressure change from its average value.

- **Period**: This is the time required for the wave to repeat itself, completing one full oscillation. Calculated as \( \frac{1}{80} \) minutes in our given problem, the period lets you find out how long each cycle lasts. Period is linked inversely to frequency; higher frequency means shorter period and more rapid oscillations.

By combining these two features, you gain a deeper understanding of how the modeled function mirrors the actual physiological process of heartbeats and blood pressure variations.