Question

After exercising for 5 min, a person has a respiratory cycle for which the rate of air flow, \(r,\) in litres per second, in the lungs is approximated by \(r=1.75 \sin \frac{\pi}{2} t,\) where \(t\) is the time, in seconds. a) Determine the time for one full respiratory cycle. b) Determine the number of cycles per minute. c) Sketch the graph of the rate of air flow function. d) Determine the rate of air flow at a time of 30 s. Interpret this answer in the context of the respiratory cycle. e) Determine the rate of air flow at a time of 7.5 s. Interpret this answer in the context of the respiratory cycle.

Short answer

The period is 4 seconds.Number of cycles per minute is 15.At 30 seconds, the rate is 0 L/s.At 7.5 seconds, the rate is -1.75 L/s.

Step-by-step solution

Determine the Period of the Function

The respiratory cycle is represented by the function \[ r = 1.75 \sin \left( \frac{\pi}{2}t \right) \].The function is a sine wave, and the general form of a sine function is given by \[ r = A \sin \left( Bt \right) \].In this case, \[ B = \frac{\pi}{2} \].The period of a sine function is given by:\[ T = \frac{2\pi}{B} \].Substitute \[ B = \frac{\pi}{2} \].\[ T = \frac{2\pi}{\frac{\pi}{2}} = 4 \text{ seconds} \].This is the time for one full respiratory cycle.

Determine the Number of Cycles Per Minute

The number of cycles per minute can be determined by dividing the number of seconds in a minute by the period of the function.\[ \text{Number of cycles per minute} = \frac{60 \text{ seconds}}{\text{period}} = \frac{60}{4} = 15 \text{ cycles per minute} \].

Sketch the Graph of the Rate of Air Flow Function

For graphing the function \( r = 1.75 \sin(\frac{\pi}{2} t) \), it follows a sine wave pattern:- The amplitude (maximum value) is 1.75.- The period is 4 seconds.- Over one period, the function starts at 0, reaches a maximum of 1.75 at 2 seconds, goes back to 0 at 4 seconds, reaches a minimum of -1.75 at 6 seconds, and returns to 0 at 8 seconds.

Determine the Rate of Air Flow at 30 Seconds

Substitute \( t = 30 \) seconds into the function \( r = 1.75 \sin(\frac{\pi}{2} t) \):\[ r = 1.75 \sin(\frac{\pi}{2} (30)) = 1.75 \sin(15\pi) = 1.75 \sin(0) = 0 \text{ litres per second} \].The interpretation here is that at 30 seconds, the rate of air flow in the lungs is zero, indicating the person is at the neutral point of the respiratory cycle (neither inhaling nor exhaling).

Determine the Rate of Air Flow at 7.5 Seconds

Substitute \( t = 7.5 \) seconds into the function \( r = 1.75 \sin(\frac{\pi}{2} t) \):\[ r = 1.75 \sin(\frac{\pi}{2} (7.5)) = 1.75 \sin(3.75\pi) \].Since \[ 3.75\pi = 3\pi + 0.75\pi \],which places us in the negative sine wave part where \[ \sin(0.75\pi) = 1 \],we get \[ r = 1.75 \sin(3.75\pi) = 1.75(-1) = -1.75 \text{ litres per second} \].This indicates that at 7.5 seconds, the rate of air flow in the lungs is -1.75 litres per second, meaning the person is exhaling at the maximum rate.

Key concepts

trigonometric functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the lengths of its sides. They are foundational in many areas of mathematics and science. The most common trigonometric functions are sine (\(\text{sin}\)), cosine (\(\text{cos}\)), and tangent (\(\text{tan}\)). These functions are periodic, meaning they repeat their values in regular intervals.
For our respiratory cycle problem, we use the sine function to model the rate of air flow in the lungs. The general form of a sine function can be written as:
\[ y = A \sin(Bt + C) + D \]
Here, \(\text{A}\) is the amplitude (the maximum value), \(\text{B}\) influences the period of the function, \(\text{C}\) shifts the graph horizontally, and \(\text{D}\) shifts it vertically.
In the given respiratory cycle function \( r = 1.75 \sin(\frac{\pi}{2}t) \), there is no horizontal or vertical shift (\(\text{C} = 0\), \(\text{D} = 0\)).

period of sine function

The period of a sine function is the interval over which the function completes one full cycle and repeats itself. For a sine function \( y = A \sin(Bt) \), the period can be found using the formula:
\[ T = \frac{2\pi}{B} \]
In our example, \(\text{B} = \frac{\pi}{2}\). To find the period:
\[ T = \frac{2\pi}{\frac{\pi}{2}} = 4 \text{seconds} \]
This means that every 4 seconds, the function \( r = 1.75 \sin(\frac{\pi}{2}t) \) completes one full cycle. Understanding the period is crucial for interpreting how frequently the events (like breaths) occur over a given time.

amplitude

Amplitude refers to the maximum value a trigonometric function can attain from its midline. In the expression \( y = A \sin(Bt) \), \(\text{A}\) represents the amplitude.
For our respiratory cycle problem, the amplitude is 1.75. This means the maximum rate of air flow is 1.75 liters per second, and the minimum rate is -1.75 liters per second. The amplitude shows how far the air flow rate oscillates from the midline of 0, giving us insight into the intensity of the breaths taken.

cycle rate

The cycle rate measures how often a repeating event happens in a specific time frame. In terms of the respiratory cycle, it provides the number of breathing cycles a person goes through per minute.
We found the period of our function to be 4 seconds earlier. To determine the number of cycles per minute:
\[ \text{Number of cycles per minute} = \frac{60 \text{ seconds}}{4 \text{ seconds per cycle}} = 15 \text{ cycles per minute} \]
This means the person breathes 15 times in one minute. The cycle rate is a crucial factor in understanding breathing patterns and overall respiratory health.

graphing sine waves

Graphing sine waves helps visualize how the function behaves over time. In our example, the function \( r = 1.75 \sin(\frac{\pi}{2}t) \) needs to be sketched to understand the pattern of air flow.
Here’s how to sketch the graph:

• Determine key points in one period (\frac{2\pi}{\text{B}} = 4 seconds).
• At \( t = 0 \), \( r = 0 \).

At its maximum:

At \( t = 2 \) seconds, \( r = 1.75 \).

Return to the midline:

At \( t = 4 \) seconds, \( r = 0 \).

Minimum value:

At \( t = 6 \) seconds, \( r = -1.75 \).

Back to the midline:

At \( t = 8 \) seconds, \( r = 0 \).

These points can be connected smoothly to create a sine wave pattern. By plotting these values, you can see the repeating structure of the respiratory cycle.