Question

1. the volume of air in a persons lungs can be mode with a periodic function. the graph below represents the volume of air, in ml, in a persons lungs over time t, measured in seconds.

what is the period and what does it represent in this context?
the period is ____ __ and it represents ____
word bank 1: (a) ml, (b) seconds
word bank 2: (a) the average air volume, (b) the time between consecutive peaks in air volume, (c) how often the breathing cycle repeats in a second, (d) the difference between maximum and minimum air volume

Explanation:

Step1: Calculate the period

The period of a periodic function is the horizontal length of one complete cycle. For a sinusoidal - like graph (which this periodic function graph resembles), if we have two consecutive peaks \((x_1,y_1)\) and \((x_2,y_2)\) (or two consecutive troughs), the period \(T=x_2 - x_1\).
We are given two consecutive peaks at \((1.25,3200)\) and \((4.25,3200)\). Then \(T = 4.25-1.25\).

$$T=4.25 - 1.25=3$$

The units of the \(x\) - axis (time \(t\)) is seconds.

Step2: Interpret the period

The period of a periodic function \(y = f(t)\) (where \(y\) is the volume of air and \(t\) is time) represents the time it takes for the function to complete one full cycle.
In the context of the volume of air in the lungs, one full cycle of the function represents one full breathing cycle (inhaling and exhaling). So, the period represents the time between consecutive peaks (or troughs) in the air - volume function, which is the time it takes to complete one full breathing cycle.

Answer:

The period is \(3\) seconds and it represents the time between consecutive peaks in air volume (or the time it takes to complete one full breathing cycle).