Problem 41 - Precalculus Mathematics for Calculus

Question

Biological Clocks Circadian rlythms are biological processes that oscillate with a period of approximately 24 hours. That is, a circadian rhythm is an internal daily biological clock. Blood pressure appears to follow such a rhythm. For a certain individual the average resting blood pressure varies from a maximum of \(100 \mathrm{mmHg}\) at 2: 00 P.M. to a minimum of \(80 \mathrm{mm} \mathrm{Hg}\) at 2: 00 A.M. Find a sine function of the form $$ f(t)=a \sin (\omega(t-c))+b $$ that models the blood pressure at time \(t,\) measured in hours from midnight.

Step-by-Step Solution

Answer

\( f(t) = 10 \sin\left(\frac{\pi}{12}(t - 6)\right) + 90 \)

Step 1: Determine the Midline, Amplitude, and Vertical Shift

The midline is the average of the maximum and minimum values. Calculate it as follows:\[ b = \frac{100 + 80}{2} = 90 \]Amplitude is half the difference between the maximum and minimum values:\[ a = \frac{100 - 80}{2} = 10 \]

Step 2: Determine the Period and Frequency

The period of a circadian rhythm is 24 hours. The standard formula for period \(T\) in terms of \(\omega\) is:\[ T = \frac{2\pi}{\omega} \]Given \(T = 24\) hours for this problem, we solve:\[ \omega = \frac{2\pi}{24} = \frac{\pi}{12} \]

Step 3: Determine the Phase Shift

The maximum occurs at 2:00 PM, which is 14 hours from midnight. The sine function typically reaches its maximum at \(\frac{\pi}{2}\), hence the phase shift \(c\) can be calculated as:\[ \omega(c) = \frac{\pi}{2} \]Replacing \(\omega\) with \(\frac{\pi}{12}\):\[ \frac{\pi}{12}(c) = \frac{\pi}{2} \]Solving for \(c\):\[ c = 6 \]So, the phase shift is 6 hours.

Step 4: Write the Sine Function

Now that we have all necessary variables, we can write the function:\[ f(t) = 10 \sin\left(\frac{\pi}{12}(t - 6)\right) + 90 \]This function models the blood pressure as a sine wave with maximum at 2:00 PM and minimum at 2:00 AM.

Key Concepts

Sine Function

The sine function is one of the foundational trigonometric functions, characterized by its smooth and symmetrical wave-like shape. It is defined mathematically as \( f(x) = a \sin(bx + c) + d \), where:

• \(a\) controls the amplitude or height of the wave.
• \(b\) affects the wave's frequency.
• \(c\) is the phase shift, which influences where the wave starts horizontally.
• \(d\) is the vertical shift, raising or lowering the entire wave.

In trigonometry, the sine function is pivotal because it models repetitive oscillating processes, such as seasonal temperatures or the alternating of day and night. Its unique qualities allow it to effectively simulate natural phenomena, especially when a regular, repeating pattern is observed.
For the exercise concerning blood pressure, the sine function \( f(t) = 10 \sin\left(\frac{\pi}{12}(t - 6)\right) + 90 \) was crafted, demonstrating how physiological cycles can be captured through mathematical functions.

Periodic Functions

Periodic functions are mathematical functions that repeat their values in regular intervals or periods. The sine function is a quintessential example of a periodic function. Its repeating cycle is denoted as its period \(T\), and for the sine function, this corresponds to the time it takes for one complete oscillation.
The key attributes of periodic functions include:

• **Period (\(T\))**: The distance or time over which the function's wave repeats itself. For the sine function in this exercise, the period is 24 hours.
• **Amplitude**: The peak value (height) from the central axis of the wave plane, which measures how far the values deviate from its mean.

In the context of biological processes, the periodic nature of the sine function effectively simulates the rhythmic variations in blood pressure, signifying why it is suitable for modeling circadian rhythms.
By understanding periodic functions, students can appreciate the predictability and constancy inherent in many natural cycles and rhythms.

Phase Shift

Phase shift in trigonometric functions refers to the horizontal displacement of the wave from its standard position. This shift is determined by the phase term \(c\) in the formula \( f(x) = a \sin(bx + c) + d \). It represents how much the wave is translated along the x-axis.
To calculate the phase shift needed to align the sine wave with real-world data, we:

• Identify where the sine function's peak should occur.
• Apply the formula \( bx + c = \frac{\pi}{2} \) for reaching the peak.
• Solve for \(c\) using known values of \(b\) and other given conditions.

In the exercise, the phase shift was calculated to align the blood pressure peak at 2:00 PM. Calculating \(c\) involved solving \( \frac{\pi}{12}(c) = \frac{\pi}{2} \), leading to \(c = 6\).
Understanding phase shift enables tuning a model to fit specific data points precisely, making it essential for accurate modeling.

Modeling Biological Processes

Modeling biological processes with mathematical functions provides a framework for understanding complex physiological behaviors. Functions like sine, known for their periodic and wave-like attributes, mirror the cyclical nature of many biological phenomena.
This approach to modeling considers:

• **Repetition**: Biological cycles, such as circadian rhythms, are inherently repetitive, making periodic functions like sine a natural choice.
• **Predictability**: Accurate models allow for the prediction of future states based on current data.
• **Accuracy**: By fitting parameters such as amplitude, period, and phase shift, the model becomes tailored to individual or precise biological data.

The specific exercise, modeling blood pressure variations, uses the sine function to replicate the daily rise and fall associated with circadian rhythms.
Integrating mathematical modeling into biology aids in planning medical treatments and understanding biological changes, enhancing our ability to anticipate and react to health variations.