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To solve part (d), we need to write a function \( V(t) = A \cos(Bt) + D \) that models the volume of air in the lungs over time \( t \), based on the given respiratory information.

### Step-by-step reasoning:

1. **Period (B)**:  
   The period \( T \) of the respiratory cycle is given as 5 seconds. The general form of the cosine function has a period of \( \frac{2\pi}{B} \). Therefore, to match the period of 5 seconds:
   \[
   \frac{2\pi}{B} = 5 \quad \Rightarrow \quad B = \frac{2\pi}{5}
   \]

2. **Amplitude (A)**:  
   The amplitude represents half the difference between the maximum and minimum values of the volume of air.  
   After inhalation, the volume is 2520 mL, and after exhalation, it is 2000 mL. Therefore:
   \[
   A = \frac{2520 - 2000}{2} = \frac{520}{2} = 260 \, \text{mL}
   \]

3. **Vertical Shift (D)**:  
   The vertical shift \( D \) represents the average value of the volume of air in the lungs:
   \[
   D = \frac{2520 + 2000}{2} = \frac{4520}{2} = 2260 \, \text{mL}
   \]

### Final function:
Using these values, the function that models the volume of air in the lungs over time is:
\[
V(t) = 260 \cos\left(\frac{2\pi}{5}t\right) + 2260
\]

Would you like any further details or clarifications?

### Related Questions:
1. How does the amplitude affect the function in a real respiratory cycle?
2. What would happen to the function if the respiratory cycle period changed?
3. Can this function be used to model other rhythmic physiological functions, and how?
4. How would you differentiate this function to find the rate of air volume change in the lungs?
5. How would you find the time when the lung volume is at its maximum?

### Tip:
When modeling periodic functions, always check the amplitude and vertical shift to ensure the function reflects the actual physical situation correctly!

Write a function V(t) = A * cos(Bt) + D to represent the volume of air in the lungs t seconds after the end of an inhalation.

This problem models the volume of air in the lungs over time using a cosine function. It explores the amplitude, period, and vertical shift of the function based on real respiratory data, with a period of 5 seconds and volume ranges between 2000 mL and 2520 mL.

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