Question

A respiratory cycle is defined as the beginning of one breath to the beginning of the next breath. The rate of air intake r (in L/sec) during a respiratory cycle for a physically fit male can be approximated by r(t) = 0.85 sin(π/4.5 t), where t is the number of seconds into the cycle. A positive value for r represents inhalation and a negative value represents exhalation. a. How long is the respiratory cycle? b. What is the maximum rate of air intake?c. Graph one cycle of thefunction. On what interval does the inhalation occur? On what interval does the exhalation occur?

Answer 1

a) t = 0,22 sec

b) 0.85 lts/sec

c) Interval for inhalation [ 0 < ( π/4.5 t) ≤ π ]

Interval for exhalation [ π < ( π/4.5 t ) ≤ 2π ]

Explanation:

The respiratory cycle is approximated

r(t) = 0.85 sin (π/4,5t)

a) how long is the respiratory cicle.

As for problem statement a respiratory cicle is time between two consecutive breath

Then cycle will go from sin (π/4.5 t) = 0 to the position when sin (π/4.5 t) becomes negative that is when sin (π/4.5 t) < 0 or just after (π/4.5 t) is equal to π , so

sin (π/4.5 t) = sin (π) or (π/4.5 t) = π

solving for t t = π /π*4,5

t = 1/ 4,5

t = 0,22 sec

b) The maximum rate of air intake is when sin( π/4.5 t) takes its maximum value which is 1.

maxmum rate of air intake is: 0.85 lts/sec

c) Interval for inhalation [ 0 < ( π/4.5 t) ≤ π ]

Interval for exhalaton [ π < ( π/4.5 t ) ≤ 2π ]