Problem 79 - Precalculus Mathematics for Calculus

Question

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 \(\mathrm{mmHg}\) (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?

Step-by-Step Solution

Answer

(a) Period: \(\frac{1}{80}\) min; (b) 80 beats/min; (d) Blood pressure: 140/90 mmHg, which is elevated.

Step 1: Find the period of the function

To find the period of the sine function in the blood pressure model, we look at the coefficient of the variable in the sine function. The model is given by: \[ p(t) = 115 + 25 \sin(160\pi t) \]In the function \(\sin(bt)\), the period \(T\) is calculated as \( T = \frac{2\pi}{b} \). Here \(b = 160\pi\). Therefore:\[ T = \frac{2\pi}{160\pi} = \frac{1}{80} \text{ minutes} \]This is the period of the sine function, representing the time it takes for one complete heartbeat cycle.

Step 2: Calculate the number of heartbeats per minute

Since the period of the heartbeat is \(\frac{1}{80}\) minutes, we calculate the number of heartbeats per minute by taking the reciprocal of the period:\[ \text{Heartbeats per minute} = \frac{1}{T} = 80 \text{ beats per minute} \]

Step 3: Derive the systolic and diastolic pressures

The maximum (systolic) and minimum (diastolic) pressures are given by the peak and trough of the sine wave.For the maximum pressure (systolic), the value of \(\sin(160\pi t)\) is 1:\[\text{Maximum } p(t) = 115 + 25 \cdot 1 = 140 \text{ mmHg}\]For the minimum pressure (diastolic), the value of \(\sin(160\pi t)\) is -1:\[\text{Minimum } p(t) = 115 + 25 \cdot (-1) = 90 \text{ mmHg}\]

Step 4: Graph the function

When graphing \(p(t)\), note that it oscillates around a midline of 115 mmHg with an amplitude of 25 mmHg. The graph will rise to a maximum of 140 mmHg and fall to a minimum of 90 mmHg, repeating every \(\frac{1}{80}\) minutes, signifying each heartbeat. The function shows a sinusoidal wave starting at 115 mmHg at \(t = 0\).

Step 5: Compare blood pressure readings to normal values

The blood pressure reading derived from \(p(t)\) is 140/90 mmHg. Compared to the normal reading of 120/80 mmHg, this reading is higher, indicating elevated blood pressure.

Key Concepts

Period of a Function

The period of a function is the length of time it takes for the function to complete one full cycle, repeating its values at regular intervals. This concept is particularly important in understanding how cyclic phenomena, such as heartbeats, can be mathematically modeled using trigonometric functions.

In trigonometric functions like sine and cosine, the period is related to the coefficient in front of the variable inside the trigonometric function. Specifically, for a sine function of the form \( \sin(bt) \), the period \( T \) is calculated as \( T = \frac{2\pi}{b} \).

In our given blood pressure model, \( p(t) = 115 + 25 \sin(160\pi t) \), the coefficient of \( t \) in the sine function is \( 160\pi \). This means:

• The period \( T \) is \( \frac{2\pi}{160\pi} \).

Simplifying this, we find \( T = \frac{1}{80} \) minutes. This indicates that it takes \( \frac{1}{80} \) of a minute for one complete heartbeat cycle to occur within this model.

Understanding the period helps us understand how frequently events, like heartbeats, occur during a specific time interval.

Heartbeats per Minute

Heartbeats per minute, often referred to as heart rate, is a crucial indicator of cardiac health and fitness. It is measured by counting the number of heartbeats that occur in one minute. In our exercise, we derive this from the period of the function modeling blood pressure.

Knowing that the period of our function \( p(t) = 115 + 25 \sin(160\pi t) \) is \( \frac{1}{80} \) minutes, we calculate the heart rate by taking the reciprocal of the period.

The formula is:

• Heartbeats per minute = \( \frac{1}{T} = 80 \text{ beats per minute.} \)

This result, 80 beats per minute, is a typical resting heart rate for a healthy adult.

Being able to determine the heart rate from a mathematical model of blood pressure is an excellent example of how mathematics intersects with healthcare, assisting in monitoring vital signs.

Blood Pressure Modeling

Blood pressure modeling using trigonometric functions helps us visualize and understand the cyclical nature of blood pressure changes during the cardiac cycle. The blood pressure model provided is:

\( p(t) = 115 + 25 \sin(160\pi t) \), where pressure \( p(t) \) is measured in \( \text{mmHg} \), and \( t \) is time in minutes.

This function signifies how blood pressure fluctuates from a baseline (also known as the midline) of 115 mmHg. It reaches a maximum of 140 mmHg and a minimum of 90 mmHg, with each wave representing one heartbeat.

The sine component \( 25 \sin(160\pi t) \) reflects the changes above and below this baseline:

• Amplitude: The distance between the midline and the peak or trough of the wave, here it is 25 mmHg.
• Systolic Pressure: Maximum pressure, calculated when \( \sin(160\pi t) = 1 \), which is 140 mmHg.
• Diastolic Pressure: Minimum pressure, calculated when \( \sin(160\pi t) = -1 \), which is 90 mmHg.

When these pressures are compared to the normal range of 120/80 mmHg, we see that they demonstrate elevated blood pressure, suggesting hypertension.

Understanding blood pressure modeling enhances the ability to interpret how pressure varies with time and can aid in diagnosing and monitoring cardiovascular conditions.