Problem 43 - Contemporary Precalculus

Question

Do parts (a) and (b) of Example 9 for a person whose blood pressure is given by $$g(t)=21 \cos (2.5 \pi t)+113$$ According to current guidelines, someone with systolic pressure above 140 or diastolic pressure above 90 has high blood pressure and should see a doctor about it. What would you advise the person in this case?

Step-by-Step Solution

Answer

Answer: Yes, the person has high blood pressure, as their diastolic pressure (minimum value) is above the threshold of 90.

Step 1: Identify the type of function given

The blood pressure function is given in the form of a cosine function: $$g(t)=21 \cos (2.5 \pi t)+113$$

Step 2: Find the amplitude, period, and phase shift of the cosine function

The function is in the form of $$g(t)=A \cos(Bt+C)+D$$, where A is the amplitude, B determines the period, C is the phase shift and D is the vertical shift. For the given function: Amplitude (A) = 21 Period = $$\frac{2 \pi}{B}$$ In this case, B = 2.5π, so the period is 2. There is no phase shift (C = 0). Vertical shift (D) = 113

Step 3: Calculate the maximum and minimum values of the function

The maximum and minimum values of a cosine function are given by (D ± A). In our case: Maximum value = D + A = 113 + 21 = 134 Minimum value = D - A = 113 - 21 = 92

Step 4: Compare the maximum and minimum values with the guidelines

According to the guidelines, the person has high blood pressure if their systolic pressure is above 140 or diastolic pressure is above 90. In our case: The maximum value (systolic pressure) is 134, which is below 140. The minimum value (diastolic pressure) is 92, which is above 90.

Step 5: Provide advice based on the comparison

As the person's diastolic pressure (minimum value) is above the threshold of 90, they have high blood pressure according to the given guidelines. It is advisable for the person to see a doctor about their blood pressure.

Key Concepts

Amplitude

Amplitude in trigonometric functions, specifically cosine and sine functions, represents the measure of the peak value of oscillation from the central axis or mean position. In simple terms, it tells us how much the function varies above and below an average value.

• In a function like \( g(t) = A \cos(Bt + C) + D \), the amplitude is the absolute value of \( A \).
• This value signifies the maximum displacement from the mean position \( D \).

For our function \( g(t) = 21 \cos(2.5 \pi t) + 113 \), the amplitude is 21. This means that, starting from the average blood pressure level of 113 (the vertical shift), the pressure will oscillate up to 21 units above or below this average, reaching highest peaks (systolic pressure) and lowest valleys (diastolic pressure).
Understanding amplitude helps determine the range and variability in contexts such as wave motion, alternating current electricity, and even in modeling physiological measurements like blood pressure variations over time.

Period

The period of a trigonometric function denotes the interval after which the function begins to repeat itself. For cosine and sine functions, the period is calculated as \( \frac{2\pi}{B} \), where \( B \) is the coefficient of \( t \) in the function. This calculation helps us understand how quickly or slowly the function cycles its pattern.

• In our example function, \( g(t) = 21 \cos(2.5 \pi t) + 113 \), \( B = 2.5\pi \).
• The period is obtained by evaluating \( \frac{2\pi}{2.5\pi} = \frac{2}{2.5} = 0.8 \).

This indicates that the blood pressure function completes one full cycle every 0.8 units of time. The periodic nature of this function efficiently models how blood pressure oscillates regularly between systolic and diastolic states, which often repeat rhythmically with the heartbeat. This concept is crucial not only in understanding repetitive biological processes but also in varied applications like signal processing and mechanical vibrations.

Systolic and Diastolic Pressure

Blood pressure readings consist of two values: the systolic and diastolic pressures, representing different states of heart function.

• Systolic pressure measures the force exerted against artery walls when the heart contracts.
• Diastolic pressure measures the pressure between heartbeats when the heart relaxes.

For our function \( g(t) = 21 \cos(2.5 \pi t) + 113 \):

• The maximum value (systolic pressure) is calculated as \( 113 + 21 = 134 \).
• The minimum value (diastolic pressure) is \( 113 - 21 = 92 \).

Although the systolic value is below the risk threshold of 140, the diastolic value being 92 is above the safety threshold of 90. This means, despite the heart's contracting pressure being within safe limits, the relaxing pressure is elevated, signaling a potential health concern. Regular monitoring and perhaps medical consultation are advised to manage the risks associated with high blood pressure, preventing possible cardiovascular complications.

Trigonometric Equations

Trigonometric equations, like the one we are examining, involve functions based on the dependability of sine and cosine rules which model periodic phenomena. These equations are vital in solving problems related to waves, oscillations, and circular movements.

• They are typically written in the form \( g(t) = A \cos(Bt + C) + D \) or with sinusoidal variants.
• Solutions to these equations require evaluations of maximum/minimum values and understanding shifts in cycles.

In the context of our blood pressure model \( g(t) = 21 \cos(2.5 \pi t) + 113 \), we observe how these equations can simulate physiological oscillations like blood pressure variations. The solutions help make crucial health assessments by comparing calculated pressure values against established health guidelines. Accurately solving trigonometric equations allows for predicting patterns and intervening when necessary, ensuring well-being in a variety of disciplines including physics, engineering, and medicine.