Problem 40 - Algebra and Trigonometry

Question

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 maximum of 100 mmHg at 2:00 P.M. to a minimum of 80 mmHg 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 Midline

The amplitude, \(a\), of the sine function represents half the difference between the maximum and minimum blood pressure values. Given max is 100 mmHg and min is 80 mmHg, the amplitude is \(a = \frac{100 - 80}{2} = 10\). The midline of the sine wave is the average of the maximum and minimum values, so \(b = \frac{100 + 80}{2} = 90\).

Step 2: Find the Period and Angular Frequency

The period of a circadian rhythm with a 24-hour cycle corresponds to the period of the sine function which is 24 hours. The angular frequency \(\omega\) is related to the period \(T\) by the formula \(\omega = \frac{2\pi}{T}\). Hence, \(\omega = \frac{2\pi}{24} = \frac{\pi}{12}\).

Step 3: Calculate the Phase Shift

The maximum blood pressure occurs at 2:00 PM, which is 14 hours past midnight. To express the sine function reaching its maximum at \(t = 14\), look at \(f(t) = a\sin(\omega(t - c)) + b\), where the phase shift \(c\) can be found from setting \(\omega(t-c) = \frac{\pi}{2}\) (since \(\sin(\frac{\pi}{2}) = 1\) for maximum). Substitute \(t = 14\) and solve for \(c\): \(\frac{\pi}{12}(14-c) = \frac{\pi}{2}\). Simplifying this gives \(14 - c = 6\), thus \(c = 8\).

Step 4: Write the Sine Function

Using all calculated values, write the function as \(f(t) = 10 \sin\left(\frac{\pi}{12}(t - 8)\right) + 90\). This function models the individual's blood pressure variation over a 24-hour period.

Key Concepts

Circadian Rhythms

Circadian rhythms are fascinating biological patterns that every human experiences. They function like internal clocks which regulate biological processes over a 24-hour period. This includes processes like sleep, metabolism, and even hormone release. You can think of them like the body's own natural way of keeping time, ensuring that various processes occur at the optimal times of the day.
These rhythms are crucial for maintaining balance in your body. For example, your blood pressure demonstrates a circadian rhythm. It fluctuates naturally over the course of a day, reaching a peak and a trough. This cycle is synchronized with external cues like sunlight, eating habits, and daily activities. However, these rhythms are internally generated and persist even in the absence of such cues.

• They help regulate sleep cycles and alertness.
• They control physiological functions like hormone levels and body temperature.
• They impact mental and physical performance.

Understanding these rhythms can drastically improve our well-being by aligning our lifestyles with our natural biological clocks.

Sine Function Modeling

Sine functions are incredibly valuable tools for modeling oscillatory systems. Just like in the given problem, sine functions are often employed to represent cyclical patterns found in nature, such as circadian rhythms. This is because sine waves naturally oscillate between a maximum and a minimum value, making them ideal for depicting repetitive variations in a system.
A sine function has various components that allow it to represent real-world patterns accurately:

• Amplitude: This represents the height of the oscillation, essentially half the distance between the maximum and minimum value. In our example, it's used to describe the variation in blood pressure.
• Midline: This is the average value around which the sine wave oscillates, akin to the average blood pressure in our problem.
• Period: The length of time it takes for the cycle to repeat. Here, it perfectly represents a 24-hour circadian cycle.
• Phase Shift: Indicates the horizontal shift of the wave, allowing the peak of the sine function to occur at a specified time, like 2 PM in the problem scenario.

By adjusting these parameters, you can tailor the sine function to fit any cyclical pattern you encounter.

Angular Frequency

Angular frequency (\( \omega \)) plays a fundamental role in understanding how quickly a trigonometric function completes its cycle. Think of it as the speed of the sine wave's oscillations. It's calculated by the formula \( \omega = \frac{2\pi}{T} \), where \( T \) is the period of the oscillation.
In the context of the exercise, the angular frequency helps to model the 24-hour circadian rhythm. With a period (\( T \)) of 24 hours, the angular frequency is \( \frac{\pi}{12} \). This means that the sine wave completes one full cycle in 24 hours, matching the natural cycle of the circadian rhythm.
Angular frequency is integral for:

• Calculating the time it takes for cyclic events to repeat.
• Determining how compressed or stretched a wave is on the time axis.
• Ensuring accuracy in modeling oscillatory behaviors like those seen in blood pressure changes.

With the right angular frequency, you can accurately harmonize mathematical models with real-world periodic behaviors, enhancing our understanding of natural patterns.