Tidal volume is the volume of air displaced in the lungs between inhalation and exhalation (the amount of air in a breath). The residual volume is given by the amount remaining in the lungs even after the deepest exhalation (i.e. the minimum amount of air remaining in the lungs).

a) Suppose that a person is running and a full breath is completed every 2 seconds. Further suppose that the person’s tidal volume during the run is 3200 mL and that the residual volume is 1150 mL. Express the volume of air in the person’s lungs in the form V(t)=Acos(Bt)+C where V is the volume in mL, t is the time in seconds, and the time t=0 corresponds to the lungs being at their fullest.

I think it may be:

V(t)=1600cos(pi x t)+1150

But a previous tutor said the "+1150" was incorrect on this website so I am not sure.

**RESOLVED**

For any future calculus students who get stumped with this (relatively simple) problem:

You must have 1150 mL of air in your lungs AT MINIMUM. That means at your local minimum in your cos graph the y-value must be 1150, which makes the last part of your equation (C) to be "+2750" not "+1150".

Therefore the final equation is:

V(t)=1600cos(pi x t)+2750

To express the volume of air in the person's lungs in the given form, we need to determine the values of A, B, and C.

Given information:
- Tidal volume: 3200 mL
- Residual volume: 1150 mL
- Full breath completed every 2 seconds

First, let's find the amplitude, A, which represents half the difference between the maximum and minimum volumes. In this case, the maximum volume corresponds to the tidal volume, and the minimum volume corresponds to the residual volume. Thus, the amplitude A is calculated as:

A = (3200 mL - 1150 mL) / 2
A = 2050 mL / 2
A = 1025 mL

Next, we need to determine the angular frequency, B, which is related to the period of the breath. The period of the breath is the time it takes to complete one full cycle, which is given as 2 seconds. The angular frequency can be calculated using the formula:

B = 2π / T
where T is the period.

B = 2π / 2
B = π rad/s

Finally, we need to find the constant term, C, which represents the displacement of the function from the zero position (when the lungs are fullest). In this case, since the lungs are fullest at t = 0, the constant term C is the tidal volume:

C = 3200 mL

Combining all these values, we can express the volume of air in the person's lungs as:

V(t) = 1025cos(πt) + 3200

So, the volume of air in the person's lungs can be represented by the equation V(t) = 1025cos(πt) + 3200, where V is the volume in mL, t is the time in seconds, and t = 0 corresponds to the lungs being at their fullest.