We know that a cosine function can model the situation. In general, the cosine function can be expressed by:
[tex]h(t)=A\cos(\omega t+\phi)+k[/tex]
Where A is the amplitude, ω is the angular frequency, φ is the phase (related to the initial value), and k is the vertical shift.
From the problem, we know that the period T is 20 seconds. We can find the angular frequency from the period using the following relation:
[tex]\begin{gathered} \omega=\frac{2\pi}{T}=\frac{2\pi}{20} \\ \\ \Rightarrow\omega=\frac{\pi}{10} \end{gathered}[/tex]
Additionally, by definition, the amplitude is half the distance from the highest to the lowest point:
[tex]\begin{gathered} A=\frac{4}{2} \\ \\ \Rightarrow A=2\text{ feet} \end{gathered}[/tex]
The average value of the cosine function is the so-called "midline", related to k:
[tex]k=10\text{ feet}[/tex]
Part B
If for t = 0, h(t) = 10 feet (the average value; the bottle is moving upwards), we can use this result to find the phase of the function:
[tex]\begin{gathered} h(0)=10 \\ 2\cos(\frac{\pi}{10}\cdot0+\phi)+10=10 \\ \\ \cos\phi=0 \\ \\ \Rightarrow\phi=-\frac{\pi}{2} \end{gathered}[/tex]
Then, the equation of the function is:
[tex]h(t)=2\cos(\frac{\pi}{10}t-\frac{\pi}{2})+10[/tex]
Part C
The graph of the function from t = 0 to when it reaches the lowest value for the first time is:
From the graph, we can see that the bottle reaches the lowest value for the first time in 15 seconds.
