Let's write a detailed step-by-step solution for determining the cosine model for the depth as a function of time.
### Step 1: Understand Given Information
- Low tide depth = 2.5 meters
- High tide depth = 5.5 meters
- Low tides occur at:
- 12:00 am (which corresponds to [tex]\( t = 0 \)[/tex] hours)
- 12:30 pm (which corresponds to [tex]\( t = 12.5 \)[/tex] hours)
- High tides occur at:
- 6:15 am (which corresponds to [tex]\( t = 6.25 \)[/tex] hours)
- 6:45 pm (which corresponds to [tex]\( t = 18.75 \)[/tex] hours)
### Step 2: Determine the Average Depth (Midline)
The average depth or midline, [tex]\( k \)[/tex], can be calculated as the average of the low tide depth and the high tide depth:
[tex]\[ k = \frac{\text{low tide depth} + \text{high tide depth}}{2} \][/tex]
[tex]\[ k = \frac{2.5 + 5.5}{2} = 4.0 \][/tex]
So, [tex]\( k = 4.0 \)[/tex] meters.
### Step 3: Determine the Amplitude
The amplitude, [tex]\( a \)[/tex], is half the difference between the high tide depth and the low tide depth:
[tex]\[ a = \frac{\text{high tide depth} - \text{low tide depth}}{2} \][/tex]
[tex]\[ a = \frac{5.5 - 2.5}{2} = 1.5 \][/tex]
So, the amplitude [tex]\( a \)[/tex] is 1.5 meters.
### Step 4: Determine the Period
The period of the tides is the time between two subsequent high tides (or low tides). In this case:
- Time difference between subsequent high tides:
[tex]\[ t_{\text{high,2}} - t_{\text{high,1}} = 18.75 - 6.25 = 12.5 \text{ hours} \][/tex]
So, the period [tex]\( P \)[/tex] is 12.5 hours.
### Step 5: Calculate the Value of [tex]\( b \)[/tex]
The value [tex]\( b \)[/tex] in the function [tex]\( d = a \cos(bt) + k \)[/tex] is calculated as:
[tex]\[ b = \frac{2\pi}{\text{period}} \][/tex]
[tex]\[ b = \frac{2\pi}{12.5} \approx 0.5026548245743669 \][/tex]
### Final Cosine Model
Given all the calculated values, we can now write the cosine model for the depth as a function of time [tex]\( t \)[/tex]:
[tex]\[ d(t) = a \cos(bt) + k \][/tex]
[tex]\[ d(t) = 1.5 \cos(0.5026548245743669 \cdot t) + 4 \][/tex]
### Summary
- Amplitude [tex]\( a \)[/tex] is: 1.5 meters
- [tex]\( b \approx 0.5026548245743669 \)[/tex]
- Average depth (midline) [tex]\( k \)[/tex] is: 4.0 meters
So, the finalized cosine model for the depth as a function of time is:
[tex]\[ d(t) = 1.5 \cos(0.5026548245743669 \cdot t) + 4.0 \][/tex]
