Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 7 s. the maximum rate of air flow into the lungs is about 0.3 l/s. this explains, in part, why the function f (t) = 3 10 sin (2πt/7) 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.

Question

Rowan Tucker

Mathematics

19 April, 21:35

Breathing is cyclic and a full respiratory cycle from the beginning of inhalation to the end of exhalation takes about 7 s. the maximum rate of air flow into the lungs is about 0.3 l/s. this explains, in part, why the function f (t) = 3 10 sin (2πt/7) 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.

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Ezekiel Kirk · Answer 1

In this problem, volume is computed:

volume = ∫ (flow rate) ·dt

... = ∫[0, t] (3/10) * sin (2πt/7) ·dt

= (3 / 10) * (7 / (2 π)) * (-cos (2 π t/7) + 1)

= (-21 / (20 π)) * (cos (2 π t / 7) - 1)

Volume is equal to 21 sin^2 (pit/7) / 10pi is the inhaled air in the lungs at time t