Question

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation (t = 0) to the end of exhalation takes about t = 3 s. the maximum rate of air flow into the lungs is about 0.5 l/s. this explains, in part, why the function f(t) = 1 2 sin(2t/3) has often been used to model the rate of air flow into the lungs. use this model to find the volume of inhaled air in the lungs at time t.

Answer 1

To calculate the inhaled volume at time t using the model f(t) = 1/2 sin(2t/3), integrate the function from 0 to t. The integrated result, V = -3/4 cos(2t/3) + 3/4, represents the total volume of air inhaled up to time t.

Step-by-step explanation:

To find the volume of inhaled air in the lungs at time t using the function f(t) = \frac{1}{2} \sin(\frac{2t}{3}), we integrate the function over the time interval of interest. The rate of air flow is modelled by the sinusoidal function, with a maximum rate of 0.5 l/s corresponding to the amplitude of the sine wave.

For a continuous function such as this, the integral from the starting time, t = 0, to any other time t gives the total volume inhaled during that time. The volume V of air inhaled from the beginning of inhalation until time t is given by:

V = ∫f(t) dt = ∫\frac{1}{2} \sin(\frac{2t}{3}) dt

The integral of \frac{1}{2} \sin(\frac{2t}{3}) from 0 to t is V = -\frac{3}{4} \cos(\frac{2t}{3})\bigg|_{0}^{t} = -\frac{3}{4} [\cos(\frac{2t}{3}) - \cos(0)]. Since \cos(0) = 1, we can simplify this to:

V = -\frac{3}{4} \cos(\frac{2t}{3}) + \frac{3}{4}

This gives us the volume of inhaled air in the lungs at any time t using the given model for the air flow rate.

Answer 2

The volume of inhaled air in the lungs at time ( t ) is given by

`\( V(t) = -(1)/(3) \cos\left((2t)/(3)\right) + C \)`,

where ( C ) is the constant of integration.

Step-by-step explanation:

The given function

`\( f(t) = (1)/(2) \sin\left((2t)/(3)\right) \)`

models the rate of air flow into the lungs. To find the volume of inhaled air, we integrate the rate function with respect to time:

`\[ V(t) = \int f(t) \, dt = \int (1)/(2) \sin\left((2t)/(3)\right) \, dt \]`

Integrating
`\( \sin\left((2t)/(3)\right) \`) involves applying the chain rule in reverse, resulting in
`\( -(1)/(3) \cos\left((2t)/(3)\right) \)`. The constant of integration ( C ) is included to account for any initial volume.

Therefore, the volume function is
`\( V(t) = -(1)/(3) \cos\left((2t)/(3)\right) + C \).`

In this context, the negative sign indicates that the volume decreases during inhalation, and the cosine function reflects the cyclic nature of the breathing process. The constant ( C ) is determined by initial conditions, such as the volume of air in the lungs at ( t = 0 ). The volume of inhaled air can be obtained by evaluating this function for specific values of ( t ).