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 June 30, 2009, high tide occurred at 6:45 am. Find a function involving the cosine function that models the water depth Dstd (in meters) as a function of time t (in hours after midnight) on that day

Answer 1

To model the water depth in the Bay of Fundy as a function of time, we can use a cosine function with a time offset of 6.75 hours. The function is Dstd = (12-2)cos((2π/12)(t - 6.75)) + 2.

Step-by-step explanation:

To model the water depth Dstd (in meters) as a function of time t (in hours after midnight) on June 30, 2009, we can use a cosine function. The cosine function represents a periodic oscillation, which aligns with the natural period of oscillation of 12 hours.

Since high tide occurred at 6:45 am, we need to account for the time offset. We can express this offset as (t - 6.75), which gives us the time in hours after high tide. Multiplying this by 2π/12 gives us the angular frequency of the oscillation.

Putting these pieces together, the function that models the water depth Dstd is: Dstd = (12-2)cos((2π/12)(t - 6.75)) + 2, where Dstd represents the water depth, t represents the time in hours after midnight, and 2 and 12 are the maximum and minimum depths respectively.

Answer 2

`y = 5cos((\pi )/(12)(t-6.75))+7`

Step-by-step explanation:

You have been asked to find the values in the function:

`y=Acos(B(t+C))+D`

First, calculate A, called the amplitude, as half the value from peak to peak (from lowtide to high tide):

`A = (12-2)/2=5`

Then, calculate the vertical shift D as the average value between the high tide and lowtide:

`D = (12+2)/2=7`

Then, if P is the natural period of the tide, you can calculate B, or the frequency as:

`B=(\pi )/(P)=(\pi )/(12)`

Finally you need to find C or the phase shift thinking that for negative values, the function is shifted (displaced) to the right.

C=-6.75