When a person is breathing normally, the amount of air in their lungs can be modeled with a sinusoidal function. When full, Karen’s lungs hold $2.8$ liters of air. When “empty,” her lungs hold $0.6$ liters of air. Her brother starts timing her breathing. At $t = 2$ seconds Karen has exhaled completely and at $t = 5$ seconds she has completely inhaled.  

1. Write an equation to model the amount of air in Karen’s lungs at any time $t$ in seconds, assuming she is breathing normally.

   Hint (a):

   Make a sketch. Include a minimum point at $\left(2, 0.6\right)$ and a maximum point at $\left(5, 2.8\right)$.

   More Help (a):

   amplitude: $0.5\left(2.8 − 0.6\right)$
   period: $2\left(5 − 2\right)$
   horizontal shift: $0.5\left(5 + 2\right)$ using sine
   vertical shift: $0.5\left(0.6 + 2.8\right)$

2. What is the amount of air in Karen’s lungs if she starts holding her breath, $3.5$ seconds into the timing?

   Hint (b):

   Let $t = 3.5$ in your equation from part (a).

3. When is the first time Karen has $2.3$ liters of air in her lungs?

   Hint (c):

   Set your equation from part (a) equal to $2.3$ and solve for $t$.