Problem 60 - Contemporary Precalculus

Question

A typical healthy person's blood pressure can be modeled by the periodic function $$f(t)=22 \cos (2.5 \pi t)+95$$ where \(t\) is time (in seconds) and \(f(t)\) is in millimeters of mercury. Which one of \(.5, .8,\) or 1 appears to be the period of this function?

Step-by-Step Solution

Answer

Answer: The period of the function is 0.8 seconds.

Step 1: Identify the parameter affecting the period of the cosine function

In the given equation, $$f(t)=22 \cos(2.5 \pi t)+95,$$ the parameter affecting the period of the cosine function is the coefficient of the \(t\) term inside the cosine function, which is \(2.5 \pi\).

Step 2: Calculate the period of the function

The period of a cosine function can be calculated using the formula $$T=\frac{2\pi}{|B|}$$, where \(T\) is the period and \(B\) is the coefficient of \(t\) in the argument of the cosine function. In this case, we have \(B=2.5 \pi\). Plugging this into the formula, we get: $$T=\frac{2\pi}{|2.5\pi|}=\frac{2}{2.5}=0.8$$

Step 3: Compare the calculated period with the given options

The calculated period of the function is \(T=0.8\). Comparing this with the given options, we can conclude that the period of this function is 0.8 seconds.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, commonly used to model periodic phenomena. These functions include sine, cosine, and tangent, among others. Each function has a unique shape and set of characteristics that make it suitable for different applications.

In the context of our example, the cosine function is used to model the periodic behavior of blood pressure over time. This periodic nature allows us to predict values at any given time point. Periodicity is a core property of trigonometric functions, meaning the values repeat at regular intervals. This makes them ideal for modeling repeating cycles like waves, vibrations, or even phases of the heart pumping blood.

Understanding trigonometric functions helps in solving real-world problems modeled by these cycles, like predicting tides, sound waves, or even day-to-day fluctuations in blood pressure.

Cosine Function

The cosine function is an essential trigonometric function characteristically used to model scenarios where waves or oscillations need to be represented. It is defined as:

• cos(x) describes the cosine of angle x
• It has a range from -1 to 1
• The graph is a smooth, continuous wave

In mathematical modeling, such as in our blood pressure example, the cosine function helps describe the fluctuations from the mean pressure accurately.

The general equation for the cosine function is:
\[ f(x) = A \cos (Bx + C) + D \]
where:

• A represents the amplitude - how high and low the peaks and troughs are
• B affects the period - how wide or narrow the waves are
• C adjusts the horizontal shift - moves the graph left or right
• D is the vertical shift - this modifies the midline up or down

In our situation, the cosine function models the blood pressure variation with each heartbeat, with parameters set to reflect the specific dynamics of blood pressure changes.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent a real-world phenomenon. It helps in understanding, predicting, and potentially controlling various scenarios by providing an abstract, simplified representation of reality. Through modeling, complex systems can be understood in a relatable way.

In our exercise, we have used a cosine curve to model blood pressure levels of a healthy person over time. By utilizing the equation:
\[ f(t) = 22 \cos (2.5 \pi t) + 95 \]
we can predict blood pressure dynamics at any given second by understanding how variables interact.

This type of modeling provides insights into:

• Predicting average and peak pressures
• Understanding time cycles of pressure changes
• Adjusting model parameters to fit individual health profiles

Overall, mathematical modeling through functions like cosine allows us to decipher patterns and gain actionable insights into biological processes, such as blood pressure regulation, leading to improved health outcomes.