The Spirogram: Rate Of Change Of Air Volume In Lungs

(SiPjAjk) = S7P5A51      Basic Sequences = 1275-5-51

Expressions Of Pj Problems
The Spirogram: Rate Of Change Of Air Volume In Lungs
Math

1. A spirogram is a device that records on a graph the volume of air in a person's lungs as a function of time. If a person undergoes spontaneous hyperventilation, the spirogram trace will closely approximate a sine curve. A typical trace is given by the following equation:
V(t) = 3 + 0.05sin(160πt - π/2), where t is time measured in minutes and V(t) is the lung volume in liters.
(a) What is the formula for the rate of flow of air into the lungs at time t?
(b) What is the peak inspiration flow, that is, the maximum rate of flow of air during inspiration (breathing in)?
(c) What is the mean inspiration flow, that is, the average rate of flow of air during inspiration, from t = 0 to t = 1/160?

1(a)The string: S7P5A51 (physical change - rate of flow )
The math: Rate of flow = dV(t)/dt = 8πcos(160πt - π/2)------(1).
(b) Peak inspiration flow occurs when equation (1) is maximum. This occurs when
cos(160πt - π/2) = 1. So, peak inspiration flow = 8πL/min.
(c) Mean inspiration flow = integration of rate of flow from t = 0 to t =1/160
=1/(1/160 -0)[∫01/160 (8πcos(160πt - π/2)]
= 160(8π/160π)[sin(160πt - π/2]01/160
= 8[sin(π - π/2) - sin(- π/2) ] = 16 L/min.

The point . is a mathematical abstraction. It has negligible size and a great sense of position. Consequently, it is front and center in abstract existential reasoning.

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