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

Mathematics

Melody Baker

3 July, 22:15

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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Shelby Potts · 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