Problem 45 - Applied Calculus

Question

A person's blood pressure, \(p,\) in millimeters of mercury \((\mathrm{mm} \mathrm{Hg})\) is given, for \(t\) in seconds, by $$p=100+20 \sin (2.5 \pi t)$$ (a) What are the maximum and minimum values of blood pressure? (b) What is the time between successive maxima? (c) Show your answers on a graph of blood pressure against time.

Step-by-Step Solution

Answer

(a) Maximum: 120 mmHg, Minimum: 80 mmHg. (b) Time between maxima: 0.8 seconds.

Step 1: Understand the Sinusoidal Function

The given function is a sinusoidal function: \(p = 100 + 20\sin(2.5\pi t)\). This can be represented as \(p = A + B\sin(Ct)\), where \(A = 100\), \(B = 20\), and \(C = 2.5\pi\). In general, the function \(A + B\sin(Ct)\) oscillates between \(A - B\) and \(A + B\).

Step 2: Calculate Maximum and Minimum Values

For a function \(A + B\sin(x)\), the maximum value occurs when \(\sin(x)\) is 1, and the minimum value occurs when \(\sin(x)\) is -1. Therefore, the maximum blood pressure is \(100 + 20 \times 1 = 120\ \mathrm{mmHg}\) and the minimum blood pressure is \(100 - 20 \times 1 = 80\ \mathrm{mmHg}\).

Step 3: Calculate the Period of the Function

The period of a sinusoidal function \(y = \sin(Ct)\) is given by \(\frac{2\pi}{C}\). Here, \(C = 2.5\pi\), so the period is \(\frac{2\pi}{2.5\pi} = \frac{2}{2.5} = 0.8\) seconds. This is the time it takes for the function to complete one full cycle.

Step 4: Determine Time Between Successive Maxima

In a sinusoidal function, the time between successive maxima (or minima) is the same as the period of the function. Therefore, the time between successive maxima is \(0.8\) seconds.

Step 5: Sketch the Graph

To sketch the graph, plot the points for one full period \(0 \le t \le 0.8\) seconds. The blood pressure starts from the base value \(100 \ \mathrm{mmHg}\), reaches a maximum of \(120 \ \mathrm{mmHg}\) at \(t = 0.2\ seconds\), returns to \(100 \ \mathrm{mmHg}\) at \(t = 0.4\ seconds\), reaches a minimum of \(80 \ \mathrm{mmHg}\) at \(t = 0.6\ seconds\), and then returns to \(100 \ \mathrm{mmHg}\) at \(t = 0.8\ seconds\). Repeat this pattern for additional cycles if needed.

Key Concepts

Understanding Sinusoidal Functions

Sinusoidal functions are mathematical models belonging to the family of trigonometric functions. These are typically represented in the form: \( y = A + B \sin(Ct) \) or \( y = A + B \cos(Ct) \). Sinusoidal functions oscillate smoothly and periodically between certain values.
Here, \( A \) is the vertical shift, signifying the midpoint around which the oscillation occurs. \( B \) represents the amplitude, which is the maximum deviation from this midpoint.
The variable \( C \), known as the angular frequency, determines how quickly the oscillations occur. When dealing with sinusoidal functions, we identify properties such as:

• Amplitude: The distance from the midpoint to the peak is \( B \). For our function \( p = 100 + 20 \sin(2.5 \pi t) \), the amplitude is \( 20 \).
• Vertical Shift: This is \( A \), equal to \( 100 \). This means the base blood pressure value or midpoint oscillates around \( 100 \, \mathrm{mmHg} \).
• Maximum and Minimum Values: Calculated using \( A \pm B \). For instance, \( 100 + 20 \) leads to a max of \( 120 \, \mathrm{mmHg} \) and \( 100 - 20 \) gives a min of \( 80 \, \mathrm{mmHg} \).

Modeling Blood Pressure with Sinusoidal Functions

Blood pressure can be modeled using sinusoidal functions to reflect its cyclic nature in the human body. The heart pumps blood periodically, and this cyclical action can be captured mathematically, often simplified by sinusoidal functions. Here's how it applies:

• Physiological Basis: Heartbeats cause pressure variations. During the cardiac cycle, blood pressure peaks during heart contraction (systole) and drops during relaxation (diastole). A sinusoidal function represents this rise and fall effectively.
• Equation Application: Our given equation \( p = 100 + 20 \sin(2.5 \pi t) \) uses these principles to model blood pressure. It reflects the heart's rhythmic action, with a maximum rise and fall from a baseline value.
  This type of function captures the idea of periodic oscillations which are akin to the cycles found in biological systems like the cardiac cycle. These concepts allow us to predict blood pressure changes over time.

Graphing Trigonometric Functions for Blood Pressure Analysis

Understanding the graphing of sinusoidal or trigonometric functions is essential when visualizing phenomena such as blood pressure. For the function \( p = 100 + 20 \sin(2.5 \pi t) \), setting up the graph involves several steps:

• Identifying Key Points: Start by noting the values for critical points over one cycle of the period calculated earlier, \( 0.8 \) seconds.
  At \( t = 0 \), pressure starts from \( 100 \, \mathrm{mmHg} \); at \( t = 0.2 \) seconds, it reaches a peak at \( 120 \, \mathrm{mmHg} \); at \( t = 0.4 \), returns to \( 100 \, \mathrm{mmHg} \); at \( t = 0.6 \), hits a minimum of \( 80 \, \mathrm{mmHg} \); and finally returns to \( 100 \, \mathrm{mmHg} \) at \( t = 0.8 \).
• Plotting the Graph: Connect these points to illustrate the sinusoidal curve. The peaks and troughs should symmetrically fluctuate around the baseline of \( 100 \, \mathrm{mmHg} \).
• Repeating Patterns: Extend the graph beyond one period to visualize how these oscillations repeat over time, essential for understanding long-term trends.

Graphing these functions gives a clear visual representation, simplifying the understanding of periodic data changes and potential predictions in a real-world context like blood pressure dynamics.