Circadian Rhythms Everybody's blood pressure varies over the...

Circadian Rhythms Everybody's blood pressure varies over the course of the day. In a certain individual the resting diastolic blood pressure at time $t$ is given by $B(t)=80+7 \sin (\pi t / 12),$ where $t$ is measured in hours since midnight and $B(t)$ in mmHg (millimeters of mercury). Find this person's resting diastolic blood pressure at (a) $6 : 00$ A.M. (b) $10 : 30$ A.M. $(\mathbf{c})$ Noon (d) $8 : 00$ P.M.

Circadian Rhythms Everybody's blood pressure
varies over the course of the day. In a certain individual
the resting diastolic blood pressure at time $t$ is given by
$B(t)=80+7 \sin (\pi t / 12),$ where $t$ is measured in hours
since midnight and $B(t)$ in mmHg (millimeters of mercury).
Find this person's resting diastolic blood pressure at
(a) $6 : 00$ A.M. (b) $10 : 30$ A.M. $(\mathbf{c})$ Noon (d) $8 : 00$ P.M.

Key Concepts

Time Variable Conversion

Time variable conversion is the process of translating real-world clock times into the units used in a mathematical model. For instance, converting standard clock times such as 6:00 A.M. or 8:00 P.M. into hours since a reference point (commonly midnight) is vital for accurately evaluating functions that depend on time. This conversion ensures correct positioning on the time axis of the model.

Sinusoidal Functions

Sinusoidal functions, which include the sine and cosine functions, are used to model periodic behavior. They have a characteristic wave-like pattern that repeats at regular intervals, making them ideal for representing phenomena that oscillate over time. Key features of these functions include amplitude (the height of the peaks), period (the time between repetitions), and vertical shift (the function’s baseline adjustment).

Circadian Rhythms

Circadian rhythms are natural, internal processes that regulate the sleep-wake cycle and repeat roughly every 24 hours. In biological contexts, many physiological parameters, such as blood pressure and body temperature, follow a circadian pattern. Modeling these rhythms with periodic functions helps in understanding and predicting daily variations in bodily functions.

Function Evaluation

Function evaluation involves substituting specific values into a mathematical function to determine the output at those points. In the context of periodic models, such as those used to represent circadian rhythms, function evaluation allows one to compute the state of the system (for example, blood pressure) at given times by plugging the corresponding time value into the sinusoidal function.

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