Problem 79 - Algebra and Trigonometry

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 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

The blood pressure is 140/90 mmHg, which is higher than the normal 120/80 mmHg, indicating hypertension.

Step 1: Identify the function form

The function given is \(p(t) = 115 + 25 \sin(160 \pi t)\). This is a sine function of the form \(A + B \sin(Ct)\). In this case, \(A = 115\), \(B = 25\), and \(C = 160 \pi\).

Step 2: Determine the period

The period of a sine function \(p(t) = A + B \sin(Ct)\) is given by \(\frac{2\pi}{C}\). Substituting \(C = 160 \pi\), the period is \(\frac{2\pi}{160 \pi} = \frac{1}{80}\) minutes.

Step 3: Calculate heartbeats per minute

Since the period is the time for one complete cycle (or heartbeat), and the period is \(\frac{1}{80}\) minutes, the number of heartbeats per minute is the reciprocal of the period: \(80\) beats per minute.

Step 4: Graph the function

To graph the function, note the midline is at 115 mmHg, the amplitude is 25 mmHg, so the function varies from 90 mmHg (\(115-25\)) to 140 mmHg (\(115+25\)). It completes one cycle every \(\frac{1}{80}\) minutes, so in one minute, there will be 80 cycles.

Step 5: Determine systolic and diastolic pressures

The systolic pressure (maximum pressure) is the maximum value of \(p(t)\) given by \(115 + 25 = 140\) mmHg, while the diastolic pressure (minimum pressure) is \(115 - 25 = 90\) mmHg.

Step 6: Compare to normal blood pressure

The blood pressure reading from the function is 140/90 mmHg. Compared to the normal blood pressure of 120/80 mmHg, this is higher, indicating hypertension.

Key Concepts

Sine Function

The sine function is a trigonometric function that is fundamental in studying periodic phenomena like sound waves and heartbeats. In the context of modeling blood pressure, the sine function is used to describe the cyclical nature of pressure changes in the heart. The general form of a sine function is given by \[ y = A + B \sin(Ct) \]where:

• \(A\) is the midline or average value around which the sine wave oscillates.
• \(B\) is the amplitude, which represents the maximum deviation from the midline.
• \(C\) influences the period, determining how quickly the function cycles.

For the given blood pressure function \(p(t) = 115 + 25 \sin(160 \pi t)\), the sine function describes the rhythmic increase and decrease in blood pressure, correlating nicely with the heart's activity cycle.

Period of a Function

The period of a function refers to the interval after which the function repeats its values. For sine functions, the period can be calculated with the formula:\[ \text{Period} = \frac{2\pi}{C} \]where \(C\) is the coefficient of \(t\) in the function.In our example, \(C = 160 \pi\), so the period is:\[ \frac{2\pi}{160\pi} = \frac{1}{80} \text{ minutes} \]
This period tells us it takes only \(\frac{1}{80}\) minutes or 0.0125 minutes for the blood pressure function to complete a full cycle. This relatively quick cycle time corresponds with the rapid beating of the heart, illustrating how closely mathematical functions can model biological processes.

Heartbeats Per Minute

Heartbeats per minute (BPM) is a common measure of heart activity. It represents the number of times your heart beats in one minute. To find the BPM from a periodic function, you can use the reciprocal of the period.With a function period of \(\frac{1}{80}\) minutes, the number of heartbeats per minute is:\[ \text{BPM} = \frac{1}{\frac{1}{80}} = 80 \text{ beats per minute} \]This means the heart cycles through its systolic and diastolic phases 80 times each minute, which is a typical BPM for a healthy adult at rest. This reinforces the accuracy of using trigonometric models to understand bodily functions.

Blood Pressure Calculation

Blood pressure is a critical indicator of cardiovascular health and is usually recorded as two values: systolic over diastolic pressure. In the blood pressure function \(p(t) = 115 + 25 \sin(160 \pi t)\), these values are found as follows:

• Systolic Pressure: The maximum value is \(115 + 25 = 140\) mmHg.
• Diastolic Pressure: The minimum value is \(115 - 25 = 90\) mmHg.

The reading of 140/90 mmHg is higher than the normal range of 120/80 mmHg, suggesting a condition known as hypertension. Hypertension can increase the risk of heart disease and requires monitoring and management. By mathematically modeling blood pressure, we obtain a clearer picture of these fluctuations over time and can assess whether interventions might be necessary to maintain healthy pressure readings.