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The depth of the water at the end of a pier changes periodically along with the movement of tides. On a particular day, low tides occur at 12:00 am and 12:30 pm, with a depth of 2.5 m, while high tides occur at 6:15 am and 6:45 pm, with a depth of 5.5 m. Let [tex]$t = 0$[/tex] be 12:00 am.

Write a cosine model, [tex]$d = a \cos(b t) + k$[/tex], for the depth as a function of time.

The amplitude is [tex]$ \frac{5.5 - 2.5}{2} = 1.5$[/tex].

[tex][tex]$a = 1.5$[/tex][/tex]

The average depth is [tex]$ \frac{5.5 + 2.5}{2} = 4.0$[/tex].

[tex]$k = 4.0$[/tex]

The period is [tex][tex]$12$[/tex][/tex] hours.

[tex]$b = \frac{2\pi}{12} = \frac{\pi}{6}$[/tex]



Answer :

GinnyAnswer
Sure, let's derive the cosine model step-by-step based on the given information about the tidal depth.

### 1. Identifying Key Parameters

- Low Tide: 12:00 am and 12:30 pm (depth = 2.5 m)
- High Tide: 6:15 am and 6:45 pm (depth = 5.5 m)
- Zero Time (t=0): 12:00 am

### 2. Calculating the Amplitude (a)
The amplitude of the cosine function represents the maximum deviation from the average depth.

[tex]\[ a = \frac{\text{Highest Depth} - \text{Lowest Depth}}{2} = \frac{5.5 - 2.5}{2} = 1.5 \text{ meters} \][/tex]

### 3. Determining the Vertical Shift (k)
The vertical shift is the average depth of the water, which is halfway between the highest and lowest depths.

[tex]\[ k = \frac{\text{Highest Depth} + \text{Lowest Depth}}{2} = \frac{5.5 + 2.5}{2} = 4.0 \text{ meters} \][/tex]

### 4. Finding the Period (T) and Frequency (b)
The period of the tide is the duration in which the depth completes a full cycle (low tide to high tide and back to low tide).

From low tide to the next low tide, it takes 12 hours (from 12:00 am to 12:30 pm) and 30 minutes, which is 12.5 hours.

[tex]\[ T = 12.5 \text{ hours} \][/tex]

The frequency [tex]\(b\)[/tex] in the cosine function is given by:

[tex]\[ b = \frac{2\pi}{T} = \frac{2\pi}{12.5} \approx 0.5027 \][/tex]

### 5. Expressing the Model
We can now write the cosine model [tex]\(d\)[/tex] as a function of time [tex]\(t\)[/tex].

[tex]\[ d(t) = a \cos(bt) + k \][/tex]

Substituting the values:

[tex]\[ d(t) = 1.5 \cos(0.5027t) + 4.0 \][/tex]

So, the complete cosine model for the depth of the water at the end of the pier as a function of time is:

[tex]\[ d = 1.5 \cos(0.50 t) + 4.0 \][/tex]

### Summary of Parameters:
- Amplitude [tex]\(a = 1.5\)[/tex]
- Vertical Shift [tex]\(k = 4.0\)[/tex]
- Period [tex]\(T = 12.5\)[/tex] hours
- Frequency [tex]\(b \approx 0.50\)[/tex]

The cosine model for the depth [tex]\(d\)[/tex] as a function of time [tex]\(t\)[/tex] is:

[tex]\[ d = 1.5 \cos(0.50 t) + 4.0 \][/tex]

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