Problem 28 - Algebra and Trigonometry

Question

Each time your heart beats, your blood pressure increases, then decreases as the heart rests between beats. A certain person's blood pressure is modeled by the function $$p(t)=115+25 \sin (160 \pi t)$$ where \(p(t)\) is the pressure in \(\mathrm{mmHg}\) at time \(t,\) measured in minutes. (a) Find the amplitude, period, and frequency of \(p .\) (b) Sketch a graph of \(p\) . (c) If a person is exercising, his heart beats faster. How does this affect the period and frequency of \(p ?\)

Step-by-Step Solution

Answer

Amplitude: 25; Period: \(\frac{1}{80}\) minutes; Frequency: 80 beats/min. During exercise, the period decreases, and frequency increases.

Step 1: Identify the amplitude

The given function for blood pressure is \( p(t) = 115 + 25 \sin(160 \pi t) \). The amplitude is the coefficient of the sine function, which is the maximum deviation from the middle value of the function.Therefore, the amplitude is \( 25 \).

Step 2: Calculate the period

The period of a sine function \( y = a \sin(bx) \) is calculated by the formula \( \frac{2\pi}{b} \). Here, \( b = 160\pi \), so the period is \[ \frac{2\pi}{160\pi} = \frac{1}{80} \text{ minutes}. \]

Step 3: Determine the frequency

Frequency is the reciprocal of the period. Once we know the period is \( \frac{1}{80} \) minutes, the frequency can be calculated as \[ 80 \text{ beats per minute}. \]

Step 4: Describe changes during exercise

When a person exercises, the heart rate increases. This means the period of the function decreases as the heart beats faster, resulting in a higher frequency. This reflects in the function by an increase in the value of \( b \) resulting in a shorter period and higher frequency.

Key Concepts

Amplitude

The amplitude of a trigonometric function is crucial in understanding how much the function moves up and down from its average value. For a sine function, like the one in our example, the amplitude is given by the multiplier in front of the sine term. In the function \( p(t) = 115 + 25 \sin(160 \pi t) \), the amplitude is 25.

This means the blood pressure varies by 25 mmHg above and below its average level. It tells us the extent of the oscillation of the pressure due to the heartbeat.

• An amplitude of 25 means the total range is 50 mmHg – 25 above the average and 25 below.
• The midpoint value around which the function oscillates is 115 mmHg.

Period

The period of a trigonometric function describes the time it takes for the function to complete one full cycle. For our example \( p(t) = 115 + 25 \sin(160 \pi t) \), we calculate the period using the formula:

\[ \text{Period} = \frac{2\pi}{b} \]

With our function, \( b = 160\pi \), so the period is:
\[ \frac{2\pi}{160\pi} = \frac{1}{80} \text{ minutes} \]

This means every 0.0125 minutes (or every 0.75 seconds), the blood pressure completes one full cycle of rising and falling, aligning with the heart's beating pattern.

• The smaller the period, the faster the cycles.
• During exercise, a smaller period indicates a faster heartbeat.

Frequency

The frequency of a trigonometric function tells us how many complete cycles happen in a given unit of time. It's calculated as the reciprocal of the period. For the function \( p(t) = 115 + 25 \sin(160 \pi t) \), once we know the period is \( \frac{1}{80} \) minutes, the frequency can be determined.

The formula is:
\[\text{Frequency} = \frac{1}{\text{Period}} \]
So, the frequency is
\[ 80 \text{ beats per minute} \].

This tells us that in one minute, the blood pressure undergoes 80 complete oscillations – thus, 80 heartbeats.

• Higher frequency means more heartbeats per minute.
• During exercise, frequency increases reflecting a faster heartbeat.

Graph of Trigonometric Functions

Graphing a trigonometric function helps visualize its behavior over time. For \( p(t) = 115 + 25 \sin(160 \pi t) \), the graph shows a sinusoidal wave that oscillates between 90 mmHg and 140 mmHg.

This range is calculated as follows: the function's amplitude of 25 means it ascends 25 mmHg above the baseline of 115 and descends 25 mmHg below that point. Here's how you visualize it:

• The graph is centered at 115 mmHg.
• It peaks at 140 mmHg and troughs at 90 mmHg.
• Each cycle of the wave occurs every \( \frac{1}{80} \) minutes.

Understanding how to sketch these graphs aids in recognizing patterns and predicting future values of periodic functions.

Sinusoidal Functions

Sinusoidal functions, such as sine and cosine, are used to model periodic phenomena, like sound waves, light waves, and even heartbeats. In our problem, the sinusoidal function \( p(t) = 115 + 25 \sin(160 \pi t) \) mimics the regular oscillation of blood pressure with each heartbeat.

These functions are defined by several key characteristics:

• **Amplitude:** Determines the height of the wave from the centerline.
• **Period:** Measures how long it takes to complete one cycle.
• **Frequency:** Indicates how many cycles occur in a given timeframe.

Involving transformations such as shifts and stretches, sinusoidal functions can adapt to various real-world applications, accurately reflecting the dynamics of the systems they model.