Problem 41 - Algebra and Trigonometry

Question

Biological Clocks Circadian rhythms 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 max- imum of 100 \(\mathrm{mmHg}\) at \(2 : 00 \mathrm{PM}\) . 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

The sine function is \(f(t) = 10 \sin\left(\frac{\pi}{12}(t - 8)\right) + 90\).

Step 1: Determine Amplitude and Vertical Shift

The amplitude \(a\) is half the difference between the maximum and minimum blood pressures. Thus, \(a = \frac{100 - 80}{2} = 10\). The vertical shift \(b\) is the average of the maximum and minimum blood pressures. So, \(b = \frac{100 + 80}{2} = 90\).

Step 2: Calculate Frequency

The period of the circadian rhythm is 24 hours, which suggests a period of 24 for the sine function. Frequency \(\omega\) is calculated as \(\omega = \frac{2\pi}{T}\) where \(T = 24\). Therefore, \(\omega = \frac{\pi}{12}\).

Step 3: Determine Phase Shift

The maximum blood pressure occurs at 2 PM, which is 14 hours from midnight. Since the standard sine function reaches its maximum at \(\frac{\pi}{2}\), we set \(\omega(t-c) = \frac{\pi}{2}\) at \(t = 14\). Solving \(\frac{\pi}{12}(14 - c) = \frac{\pi}{2}\), we find \(c = 8\).

Step 4: Write the Sine Function

Substitute the values \(a = 10\), \(\omega = \frac{\pi}{12}\), \(c = 8\), and \(b = 90\) into the equation \(f(t) = a \sin(\omega(t-c)) + b\). The function becomes \(f(t) = 10 \sin\left(\frac{\pi}{12}(t - 8)\right) + 90\).

Key Concepts

Sine Function

The sine function is a fundamental concept in trigonometry and appears frequently in various real-world applications, including modeling periodic phenomena like circadian rhythms. A standard sine function takes the form \( y = a \sin(bx + c) + d \). In this formula:

• \(a\) represents the amplitude, which indicates the peak value of the wave from its center line.
• \(b\) is related to the frequency, which determines how many cycles occur in a given period.
• \(c\) is the phase shift, which moves the graph horizontally along the x-axis.
• \(d\) is the vertical shift, which moves the graph up or down along the y-axis.

For the example of modeling blood pressure in circadian rhythms, the sine function is adjusted to fit the data of maximum and minimum blood pressure values. This involves calculating specific parameters like amplitude and frequency to accurately mimic the biological cycle.

Circadian Rhythm

Circadian rhythms are natural, internal processes that regulate the sleep-wake cycle and repeat roughly every 24 hours. These rhythms are crucial for maintaining bodily functions such as hormone release, eating habits and digestion, body temperature, and others. They are influenced by environmental cues such as light and temperature, but they fundamentally operate on an intrinsic circadian clock. When modeling biological phenomena like blood pressure using a sine function, the 24-hour cycle of the circadian rhythm establishes the period of the function. This ensures that the model accurately reflects the natural daily fluctuations in blood pressure associated with this biological clock. The importance of understanding circadian rhythms comes into play when looking at variations in heart rate, alertness, performance, and various physiological factors that align with these cycles.

Phase Shift

The phase shift in a sine function represents the horizontal movement of the graph along the x-axis. It determines where the function starts its cycle in relation to a standard sine wave, which has its peak at \(x = \frac{\pi}{2}\). In the context of circadian rhythms, the phase shift allows us to align the sine wave with the specific timing of biological events, such as the peak of blood pressure measured at 2 PM in the given problem.Calculating the phase shift involves setting the expression \(\omega(t-c)\) equal to the angle where the maximum value occurs, typically \(\frac{\pi}{2}\) for a sine wave. This horizontal translation is crucial for ensuring that the model fits the observed timings of these biological processes accurately. By understanding and adjusting the phase shift, we can tailor trigonometric models to more precisely represent real-world periodic events.