# find the volume of inhaled air in the lungs at time t.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...

find the volume of inhaled air in the lungs at time t.

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 0.4 L/s. This explains, in part, why the function

f(t) = 2/5 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.

v(t)=___________________?

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### 1 Answer

You can use this function and integrate it to find the amount of volume of inhaled air inside the lungs.

Let's assume at t = 0, the volume of inhaled air is 0 L.

`f(t) = 2/5sin((2pit)/7)`

`V = int_0^tf(t)dt`

`V = int_0^t2/5sin((2pit)/7) dt`

`V = 2/5int_0^tsin((2pit)/7) dt`

`intsin((2pit)/7)dt = -cos((2pit)/7)/((2pi)/7)`

`intsin((2pit)/7)dt = -(7/(2pi))cos((2pit)/7)`

Therefore,

`V = 2/5int_0^tsin((2pit)/7) dt = -(7/(2pi))[cos((2pit)/7) - cos(0)]`

`V = -(7/(2pi))[cos((2pit)/7) - 1]`

`V = (7/(2pi))[1-cos((2pit)/7)]`