A simple model for the flow of air in and out of the lungs of a certain mammal is given by the following equation, where $$ \mathrm{V}(\mathrm{t}) $$ (measured in liters) is the volume of air in the lungs at time $$ t \geq 0, t $$ is measured in seconds, and $$ t=0 $$ corresponds to a time at which the lungs are full and exhalation begins. Only a fraction of the eir in the lungs is exchanged with each breath. The amount that is oxchanged is calted the tidal volume. Complete parts a through $$ \mathbf{c} $$ below. $$ V^{\prime}(t)=-\frac{\pi}{6} \sin \left(\frac{\pi t}{6}\right) $$ a. Find the volume function V , assuming that $$ \mathrm{V}(0)=6 \mathrm{~L} $$. Notice that V changes over time at a known rate, $$ \mathrm{V}^{\prime} $$. Which equation below correctly gives the volume function? A. $$ V(0)=V(t)+\int_{0}^{1} V^{\prime}(x) d x $$ B. $$ V(t)=V(0)+\int_{0}^{1} V^{\prime}(x) d x $$. c. $$ V(t)=V(0)+\int_{a}^{b} V^{\prime}(t) d t $$ D. $$ V(0)=V(t)+\int_{a}^{b} V^{\prime}(t) d t $$. Find the volume function $$ V $$, assuming that $$ V(0)=6 \mathrm{~L} $$. $$ V(t)=5+\cos \left(\frac{\pi t}{6}\right) \text { (Type an exact answer.) } $$ b. What is the breathing rate in breaths/minute? The breathing rate is 5 breaths/minute. (Type an integer or a simplified fraction.) c. What is the tidal volume and what is the total capacity of the lungs? The tidal volume is $$ \square $$ liter(s). (Type an integer or a simplified fraction.)

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To solve part **c** of the problem, we need to determine both the **tidal volume** and the **total capacity of the lungs** based on the volume function $$ V(t) = 5 + \cos\left(\frac{\pi t}{6}\right) $$.

### Tidal Volume

**Tidal Volume** is the amount of air exchanged during a single breath (inhalation or exhalation).

1. **Maximum Volume $$ V_{\text{max}} $$:**
   $$
   V_{\text{max}} = 5 + \cos(0) = 5 + 1 = 6 \text{ liters}
   $$

2. **Minimum Volume $$ V_{\text{min}} $$:**
   $$
   V_{\text{min}} = 5 + \cos\left(\pi\right) = 5 - 1 = 4 \text{ liters}
   $$

3. **Tidal Volume Calculation:**
   $$
   \text{Tidal Volume} = V_{\text{max}} - V_{\text{min}} = 6 \text{ L} - 4 \text{ L} = 2 \text{ liters}
   $$

### Total Capacity of the Lungs

**Total Capacity** refers to the maximum volume of air the lungs can hold.

From the above calculations:
$$
\text{Total Capacity} = V_{\text{max}} = 6 \text{ liters}
$$

### Final Answer

- **Tidal Volume:** $$ 2 $$ liter(s)
- **Total Capacity of the Lungs:** $$ 6 $$ liter(s)

**Answer:**

Problem c Answer:

- The tidal volume is 2 liter(s).
- The total capacity of the lungs is 6 liter(s).
- **Tidal Volume:** 2 liters - **Total Capacity of the Lungs:** 6 liters

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