Exercise 1.3.29

Question

Some of the highest tides in the world occur in the Bay of Fundy on the Atlantic Coast of Canada.
At Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m. The natural period
of oscillation is about 12 hours and on a particular day, high tide occurred at 6:45 am. Find a function involving the
cosine function that models the water depth D(t) (in meters) as a function of time t (in hours after midnight) on that
day.

Khagan Gulusoy · Accepted Answer

\begin{itemize}
    \item 	extit{(amplitude)} We find the amplitude as follows: 
    $$\frac{1}{2}\cdot (12-2)=\frac{1}{2}\cdot 10=5.$$
    \item 	extit{(period)} Since the period of the cosine function is $2\pi$ and the period of oscillation is about 12 hours, the horizontal stretching factor of the desired function $D$ is $\dfrac{2\pi}{12}=\dfrac{\pi}{6}.$
    \item 	extit{(shifting upward factor)} Since the middle output value of the cosine function we did not shift up yet is 0 while the middle output value of the desired cosine function is $2+(12-2)/2=7,$ the shifting upward factor of the function $D$ is equal to $7.$ 
\end{itemize}
 Since the maximum of the cosine function $D$ is equal to $6.75$ (6.45 AM = 6.75 h), we define the function $D$ as a function of time (in hours after midnight) $t$ on that day as follows:
$$\forall t\in [0,24]:\quad D(t)=7+5\cos \left(\frac{\pi}{6}\left(t-6.75ight)ight).$$

Exercise 1.3.29

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Exercise 1.3.29

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