Question

Lung Volume Normal resting lung volume \(V\), in mL, for adult men varies over the breathing cycle and can be approximated by the model $$ v(t)=250 \sin \left[\frac{2 \pi(t-1.25)}{5}\right]+2650 $$ where \(t\) is the number of seconds after breathing begins. Use the model to estimate the volume of air in a man's lungs after 2.5 seconds, 10 seconds, and 17 seconds.

Short answer

At \(t = 2.5\) seconds, the volume is 2900 mL. At \(t = 10\) seconds, it is 2400 mL. At \(t = 17\) seconds, it is also 2400 mL.

Step-by-step solution

Understanding the Given Model

The lung volume model given is \( V(t) = 250 \, \sin \left[ \frac{2 \pi(t-1.25)}{5} \right] + 2650 \). Here, \(t\) represents the time in seconds after breathing begins and \(V(t)\) gives the lung volume at that time in mL.

Substitute\(t\)=2.5 into the Model

To find the lung volume at \(t = 2.5\) seconds, substitute \(2.5\) in place of \(t\): \[ V(2.5) = 250 \, \sin \left[ \frac{2 \pi(2.5-1.25)}{5} \right] + 2650 \].

Calculate the Sine Function for \(t\)=2.5

Perform the operations inside the sine function first: \[ \frac{2 \pi(2.5-1.25)}{5} = \frac{2 \pi(1.25)}{5} = \frac{2.5 \pi}{5} = 0.5 \pi \].Now, compute the sine of this value: \( \sin(0.5 \pi) = 1 \).

Determine Lung Volume at \(t\)=2.5

Substitute \( \sin(0.5 \pi) = 1 \) back into the equation: \[ V(2.5) = 250 \times 1 + 2650 = 2900 \text{ mL} \].

Substitute\(t\)=10 into the Model

Now repeat the process for \(t = 10\) seconds: \[ V(10) = 250 \, \sin \left[ \frac{2 \pi(10-1.25)}{5} \right] + 2650 \].

Calculate the Sine Function for \(t\)=10

Perform the operations inside the sine function first: \[ \frac{2 \pi(10-1.25)}{5} = \frac{17.5 \pi}{5} = 3.5 \pi \].Now compute the sine of this value: \( \sin(3.5 \pi) = -1 \) (since sine function is periodic with period \(2\pi\)).

Determine Lung Volume at \(t\)=10

Substitute \( \sin(3.5 \pi) = -1 \) back into the equation: \[ V(10) = 250 \times (-1) + 2650 = 2400 \text{ mL} \].

Substitute \(t\)=17 into the Model

Next, find the lung volume for \(t = 17\) seconds: \[ V(17) = 250 \, \sin \left[ \frac{2 \pi(17-1.25)}{5} \right] + 2650 \].

Calculate the Sine Function for \(t\)=17

Perform the operations inside the sine function first: \[ \frac{2 \pi(17-1.25)}{5} = \frac{31.5 \pi}{5} = 6.3 \pi \].Now compute the sine of this value: \( \sin(6.3 \pi) = -1 \) (since sine function is periodic with period \(2\pi\)).

Determine Lung Volume at \(t\)=17

Substitute \( \sin(6.3 \pi) = -1 \) back into the equation: \[ V(17) = 250 \times (-1) + 2650 = 2400 \text{ mL} \].

Key concepts

Trigonometric Functions

Trigonometric functions play a crucial role in understanding periodic phenomena, such as the variation in lung volume during breathing. The primary trigonometric functions include sine, cosine, and tangent, each of which is a ratio of sides in a right triangle. These functions are named due to their historical connection with triangles and circles. In the context of the lung volume problem, the sine function is used to model the cyclical nature of lung expansion and contraction, highlighting its usefulness in periodic phenomena.

Sine Function

The sine function, denoted as sin(θ), is particularly important for modeling periodic behaviors. It oscillates between -1 and 1 and has a period of 2π, meaning it repeats its values every 2π radians. In the lung volume model, the sine function represents the rhythmic pattern of breathing. For example, the formula given in the model, 250 sin[(2π(t-1.25)/5)] + 2650, uses the sine function to account for regular increases and decreases in lung volume, peaking and dipping at specific intervals.

Mathematical Modeling

Mathematical modeling involves using mathematical expressions to represent real-world phenomena. In our lung volume problem, the model V(t) = 250 sin[(2π(t-1.25)/5)] + 2650 serves to approximate the lung volume over time. The constants 250 and 2650 adjust the amplitude and baseline volume respectively, while the sine function captures the periodic nature of breathing. This approach allows us to predict lung volume at any given time with reasonable accuracy.

Periodic Functions

Periodic functions, such as the sine function, are essential for modeling cycles and repeated patterns. These functions repeat their values at regular intervals, called periods. In the lung volume model, the periodic function is 250 sin[(2π(t-1.25)/5)] + 2650, with a period set by the parameter within the sine function. Understanding periodic functions helps in studying phenomena that recur over time, such as breathing, seasonal weather changes, and even economic cycles.