To write the cosine model, [tex]\( d = a \cos(b t) + k \)[/tex], for the depth as a function of time, we need to determine the parameters [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( k \)[/tex]. Here's a step-by-step solution:
1. Amplitude [tex]\( a \)[/tex]:
The amplitude is the distance from the average depth (mean value of low and high tides) to either the high or low tide depth.
[tex]\[
a = \frac{\text{High tide depth} - \text{Low tide depth}}{2}
\][/tex]
Given:
- High tide depth = 5.5 meters
- Low tide depth = 2.5 meters
[tex]\[
a = \frac{5.5 - 2.5}{2} = \frac{3.0}{2} = 1.5 \text{ meters}
\][/tex]
Therefore, the amplitude [tex]\( a \)[/tex] is [tex]\( 1.5 \)[/tex] meters.
2. The vertical shift [tex]\( k \)[/tex]:
The vertical shift [tex]\( k \)[/tex] is the average value of the high and low tides.
[tex]\[
k = \frac{\text{High tide depth} + \text{Low tide depth}}{2}
\][/tex]
[tex]\[
k = \frac{5.5 + 2.5}{2} = \frac{8.0}{2} = 4.0 \text{ meters}
\][/tex]
Therefore, the vertical shift [tex]\( k \)[/tex] is [tex]\( 4.0 \)[/tex] meters.
3. The period [tex]\( T \)[/tex] and the frequency [tex]\( b \)[/tex]:
The period [tex]\( T \)[/tex] is the time it takes for the tide to complete one full cycle, from high tide to high tide or low tide to low tide. Given that high tides occur every 12 hours and 30 minutes:
[tex]\[
T = 12.5 \text{ hours}
\][/tex]
The frequency [tex]\( b \)[/tex] in the cosine function is related to the period by the formula:
[tex]\[
b = \frac{2\pi}{T}
\][/tex]
[tex]\[
b = \frac{2\pi}{12.5} = \frac{2\pi}{25/2} = \frac{4\pi}{25}
\][/tex]
Therefore, combining all these values, the cosine model [tex]\( d = a \cos(b t) + k \)[/tex] is:
[tex]\[
d(t) = 1.5 \cos\left(\frac{4\pi}{25} t\right) + 4.0
\][/tex]
