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 \(D\left( t \right)\) (in meters) as a function of time t (in hours after midnight) on that day

Short answer

The models for the water depth as a function of time \(t\) is \(D\left( t \right) = 5\cos \left( {\frac{\pi }{6}\left( {t - 6.75} \right)} \right)\).

Step-by-step solution

Water depth as a function of time t 

It is given that,at Hopewell Cape the water depth at low tide is about 2.0 m and at high tide it is about 12.0 m, sothe amplitude of the cosine function is represented as \(\frac{{12 - 2}}{2} = 5{\mathop{\rm m}\nolimits} \).

So, the average magnitude is \(\frac{{12 + 2}}{2} = 7{\mathop{\rm m}\nolimits} \).

And, period 12 hours can be used to model the water depth \(D\left( t \right)\).

High tide took place at time 6:45 AM (\(t = 6.75{\mathop{\rm h}\nolimits} \)).

Therefore, the curve begins a cycle at the time \(t = 6.75{\mathop{\rm h}\nolimits} \) (shift 6.75 units to the right).

Construct the cosine function

The function involving cosine function that models the water depth is shown below:

\(\begin{aligned}{c}D\left( t \right) &= 5\cos \left( {\frac{{2\pi }}{{12}}\left( {t - 6.75} \right)} \right) + 7\\ &= 5\cos \left( {\frac{\pi }{6}\left( {t - 6.75} \right)} \right)\end{aligned}\)

Here, \(t\) is the number of hours after midnight and \(D\) is in meters.

Thus, the function involving the cosine function that models the water depth as a function of time \(t\) is \(D\left( t \right) = 5\cos \left( {\frac{\pi }{6}\left( {t - 6.75} \right)} \right)\).