Having a lot of trouble with this one:

Q) 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 remaining in the lungs even after the deepest exhalation (ie the minimum amount of air remaining in the lungs).

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) = A cos (Bt) + C where V is the volume in millilitres, t is the time in seconds, and the time t = 0 corresponds to the lungs being at their fullest.

period of sin(kx) or cos(kx) is 2π/k.

SO, for a period of 2 seconds, k = π.

V(t) = A cos(πt)+C

residual volume is 1150, so

V(t) = A cos(πt) + 1150

Tidal volume is 3200, so

V(t) = 1600 cos(πt) + 1150

Since cos has max at t=0, we are done.

Oops. Do you see my mistake?

Hint: C is wrong.

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

Let's break down the information given:

1. Tidal volume during the run: 3200 mL
This means that the volume of air in the person's lungs varies between 3200 mL and another value during the breathing cycle.

2. Residual volume: 1150 mL
This is the minimum amount of air remaining in the lungs even after the deepest exhalation.

3. A full breath is completed every 2 seconds.
This indicates that the breathing cycle has a period of 2 seconds.

To express the volume of air in the person's lungs as V(t) = A * cos(Bt) + C, we need to find the appropriate values for A, B, and C based on the given information.

First, let's find A, which represents half the total amplitude of the volume change in the lungs. In this case, the total amplitude is the difference between tidal volume and residual volume divided by 2:

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

Next, let's find B, which corresponds to the frequency of the oscillation. Since a full breath is completed every 2 seconds, the period of the oscillation is 2 seconds. The frequency (B) can be found using the formula:

B = 2π / period

B = 2π / 2
B = π

Finally, let's find C, which represents the average volume during the breathing cycle. To calculate this, we can take the average between the tidal volume and residual volume:

C = (3200 mL + 1150 mL) / 2
C = 4350 mL / 2
C = 2175 mL

Therefore, the volume of air in the person's lungs during the run can be expressed as:

V(t) = 1025 * cos(πt) + 2175

Note: Make sure to convert the time t to seconds when evaluating the equation.