Problem 65 - Calculus Early Transcendentals

Question

A simple model (with different parameters for different people) for the flow of air in and out of the lungs is $$V^{\prime}(t)=-\frac{\pi}{2} \sin \frac{\pi t}{2}$$ where \(V(t)\) (measured in liters) is the volume of air in the lungs at time \(t \geq 0, t\) is measured in seconds, and \(t=0\) corresponds to a time at which the lungs are full and exhalation begins. Only a fraction of the air in the lungs in exchanged with each breath. The amount that is exchanged is called the tidal volume. a. Find and graph the volume function \(V\) assuming that $$ V(0)=6 \mathrm{L} $$ b. What is the breathing rate in breaths/min? c. What is the tidal volume and what is the total capacity of the lungs?

Step-by-Step Solution

Answer

Answer: The tidal volume is 1 L and the total capacity of the lungs is 6 L.

Step 1: Find the volume function V(t)

To find the volume function \(V(t)\), we need to integrate the given rate of change of volume function \(V'(t)\). $$ V'(t) = -\frac{\pi}{2} \sin\frac{\pi t}{2} $$ Integrating with respect to \(t\): $$ V(t) = \int -\frac{\pi}{2} \sin\frac{\pi t}{2} dt $$ Using the substitution \(u=\frac{\pi t}{2}\) and \(du=\frac{\pi}{2} dt\): $$ V(t) = \int -\sin u du = \cos u + C = \cos\frac{\pi t}{2} + C $$ Since \(V(0) = 6\), we can solve for \(C\): $$ 6 = \cos\frac{\pi (0)}{2} + C \implies C=5 $$ Therefore, the volume function is: $$ V(t) = \cos\frac{\pi t}{2} + 5 $$

Step 2: Graph the volume function

Now that we have the volume function \(V(t)\), we can graph it. The function is a cosine function with a period of \(4\) seconds, so it would look like a cosine wave starting from its maximum value, oscillating between \(5\) and \(6\) liters.

Step 3: Find the breathing rate

To find the breathing rate, we need to determine the time it takes for one complete breath (inhale and exhale). Since the period of the volume function is \(4\) seconds, it takes \(4\) seconds for one complete breath. Therefore, the breathing rate in breaths/minute is: $$ \text{Breathing rate} = \frac{60\,\text{seconds}}{4\,\text{seconds/breath}} = 15\,\text{breaths/min} $$

Step 4: Find the tidal volume and total capacity of the lungs

The tidal volume is the amount of air exchanged with each breath, which can be found by taking the maximum value of \(V(t)\), which is \(6\) liters, and subtracting the minimum value of \(V(t)\), which is \(5\) liters. $$ \text{Tidal volume} = 6 - 5 = 1\,\text{L} $$ The total capacity of the lungs is the maximum volume of air that the lungs can hold, which is the same as the maximum value of \(V(t)\): $$ \text{Total capacity} = 6\,\text{L} $$ In conclusion, the volume function is \(V(t) = \cos\frac{\pi t}{2} + 5\), the breathing rate is \(15\) breaths/min, the tidal volume is \(1\) L, and the total capacity of the lungs is \(6\) L.

Key Concepts

Volume Function

The volume function describes how the volume of air in the lungs changes over time. For this exercise, the rate at which air flows in and out of the lungs is given by the derivative function \( V'(t) = -\frac{\pi}{2} \sin\frac{\pi t}{2} \).
To find the actual volume function \( V(t) \), we need to integrate this derivative. By integrating, we find our volume function: \[V(t) = \cos\frac{\pi t}{2} + 5\]This equation tells us how the air volume changes cyclically over time, resembling a wave. The lungs oscillate between a minimum volume at the troughs and a maximum volume at the peaks.

• At \( t = 0 \), the value is \( 6 \) liters, indicating the lungs are full.
• The volume decreases to a minimum of \( 5 \) liters as the lungs expel air.

You can imagine this as a continuous cycle of breathing where the volume of air follows a smooth cosine curve.

Breathing Rate

The breathing rate is essentially how many complete breaths a person takes per minute. By understanding the period of our volume function, we can find this rate. Here, the period of the function is calculated using the cosine wave, which completes one cycle in 4 seconds.
Since there are 60 seconds in a minute, we can find the breathing rate like this:

• The period of the function is 4 seconds, meaning one complete inhalation and exhalation takes 4 seconds.
• Hence, a breathing rate of \(\frac{60}{4} = 15\) breaths per minute is established.

This gives us an average idea of how many breaths occur within a specific timeframe and is essential in evaluating the respiratory efficiency of a person under this model.

Tidal Volume

Tidal volume measures how much air a person breathes in or out with each cycle.
For this exercise, after determining the volume function, we can find the tidal volume by calculating the difference between the maximum and minimum volume of air in the lungs during one cycle.

• From the volume function \( V(t) = \cos\frac{\pi t}{2} + 5 \), the maximum volume is 6 liters, and the minimum is 5 liters.
• This difference, \( 6 - 5 = 1 \) liter, represents the tidal volume.

Tidal volume is crucial in understanding how much air is regularly exchanged in the lungs with each breath, helping assess lung function efficiency in different individuals and scenarios.

Lungs Capacity

The capacity of the lungs refers to the maximum volume of air the lungs can hold, which can be derived from the volume function. In this context, this maximum capacity is observed at the peaks of our cosine wave.
For our model, the total lung capacity is given by the maximum value of the function:

• The peak volume, as calculated, is 6 liters, which is the value when the lungs are fully inhaled.

This measure is essential as it provides information on the potential upper limit of air volume the lungs can manage. It helps in monitoring respiratory health, defining constraints for exercise, and diagnosing potential respiratory issues.