Problem 10 - Applied Calculus

Question

A person breathes in and out every three seconds. The volume of air in the person's lungs varies between a minimum of 2 liters and a maximum of 4 liters. Which of the following is the best formula for the volume of air in the person's lungs as a function of time? (a) \(y=2+2 \sin \left(\frac{\pi}{3} t\right) \quad\) (b) \(\quad y=3+\sin \left(\frac{2 \pi}{3} t\right)\) (c) \(y=2+2 \sin \left(\frac{2 \pi}{3} t\right)\) (d) \(y=3+\sin \left(\frac{\pi}{3} t\right)\)

Step-by-Step Solution

Answer

The best formula is option (d): \(y=3+\sin\left(\frac{2 \pi}{3}t\right)\).

Step 1: Determine Amplitude and Midline

To find the best formula, analyze the range of the lung volume, which varies from 2 liters to 4 liters. The amplitude is half of the difference in range, \[ \text{Amplitude} = \frac{4 - 2}{2} = 1 \text{ liter} \]. The midline of this oscillation is at the average volume, calculated as \[ \text{Midline} = \frac{4 + 2}{2} = 3 \text{ liters} \].

Step 2: Determine Period and Frequency

The person’s breathing cycle is every 3 seconds. This means the period of the sinusoidal function is 3 seconds. The period \(T\) of the sinusoidal function is related to the frequency \(b\) as \[ T = \frac{2\pi}{b} \]. Solving for frequency gives us \[ b = \frac{2\pi}{3} \].

Step 3: Write the Equation

Now, compile this information to write the sinusoidal function. Since the midline is 3 liters and the amplitude is 1 liter, use the form \[ y = 3 + 1\sin\left(\frac{2\pi}{3}t\right) \].

Step 4: Compare with Options

Compare the derived equation with the given options. It matches exactly with option (d): \[ y = 3 + \sin\left(\frac{2\pi}{3}t\right) \].

Key Concepts

Amplitude

Amplitude is a key feature of a sinusoidal function that describes how far the function moves above and below its midline. In simple terms, it is the maximum height of the wave from its central axis.
In the given exercise, the amplitude can be determined by looking at the variation in the lung volume, which swings between 2 liters and 4 liters. To find the amplitude, calculate half the difference between the maximum and minimum volume:

• Maximum volume = 4 liters
• Minimum volume = 2 liters
• Amplitude = \( \frac{4 - 2}{2} = 1 \text{ liter} \)

This means that the volume of air in the lungs fluctuates by 1 liter above and below the midline, which is the average point between the maximum and minimum values. Amplitude provides us with information about the range of change around the average, giving us a sense of the dynamics of the sinusoidal pattern.

Frequency

Frequency in sinusoidal functions reflects how often the cycle of the function repeats over a unit period. It is closely tied to the function's period, as it indicates the number of complete oscillations within a given timeframe.
In the exercise, the person breathes in and out every 3 seconds. This scenario defines the period, and hence influences the frequency of lungs' volume oscillation:

• Given period \( T = 3 \text{ seconds} \)
• Frequency relationship \( T = \frac{2\pi}{b} \)

Thus, solving for frequency "b" gives:

• \( b = \frac{2\pi}{3} \)

This implies that the lungs complete one cycle of breathing within three seconds. The frequency is crucial in determining how rapidly the sinusoidal function completes its cycles, impacting the smooth oscillation portrayed by the function.

Period

The period of a sinusoidal function is the duration it takes to complete one full cycle of its pattern. It is the length of time between identical points on consecutive cycles of the wave.
In this breathing exercise, the period is specified as the time taken to complete one in-and-out breathing cycle:

• Period "T" is \( 3 \text{ seconds} \).

Understanding period helps explain how long the oscillation process takes before repeating its cycle. To establish the period mathematically, use the connection with frequency:

• \( T = \frac{2\pi}{b} \)
• Thus, \( T = 3 \) implies \( b = \frac{2\pi}{3} \)

This tells us that every 3 seconds, the cycle of lung volume oscillation repeats, providing a predictable rhythm. The period is fundamental to charting the progression and timing of the sinusoidal function, enlightening us on the pace at which changes occur.