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]
