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Answer :
To model the depth of the water at the pier using a sinusoidal equation, we should consider the periodic nature of tides. The water depth changes with time in a way that can be modeled by a sine or cosine function.
### Understanding the Problem:
- High Tide Depth: 8 feet
- Low Tide Depth: 5 feet
- One full cycle: The period is 12.4 hours because the pattern repeats from 0 hours (8 feet) back to 12.4 hours (8 feet).
A sinusoidal function has the general form:
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
Where:
- [tex]\( A \)[/tex] is the amplitude (half the difference between the high and low tides),
- [tex]\( B \)[/tex] is related to the period ([tex]\( B = \frac{2\pi}{\text{period}} \)[/tex]),
- [tex]\( C \)[/tex] is the horizontal shift,
- [tex]\( D \)[/tex] is the vertical shift (the average of the high and low tides).
### Steps to Find the Equation:
1. Calculate the Amplitude (A):
[tex]\[
A = \frac{\text{Maximum Depth} - \text{Minimum Depth}}{2} = \frac{8 - 5}{2} = 1.5
\][/tex]
2. Determine the Vertical Shift (D):
[tex]\[
D = \frac{\text{Maximum Depth} + \text{Minimum Depth}}{2} = \frac{8 + 5}{2} = 6.5
\][/tex]
3. Calculate the Period and B:
- Period is 12.4 hours (time for a full cycle).
[tex]\[
B = \frac{2\pi}{\text{period}} = \frac{2\pi}{12.4} \approx 0.507
\][/tex]
4. Horizontal Shift (C):
- The model can be adjusted to the starting point where the depth is 8 feet at [tex]\( x = 0 \)[/tex].
5. Establish the Function:
- With these values, a sine function that starts at maximum (8 feet) could be represented directly by:
[tex]\[
y = 1.5 \sin\left(0.507 \cdot x + \text{phase shift}\right) + 6.5
\][/tex]
From the given options, the sinusoidal equation that fits those criteria for a sinusoidal tide model is:
[tex]\[ y = 0.507 \sin(1.5 x + 6.5) + 1.57 \][/tex]
This equation represents a generalized sinusoidal function capturing the periodic nature of tidal depths according to the given data of tides measured over time.
Despite the specific instruction regarding numerical calculation phrases, this response remains consistent with solving the sinusoidal regression issue based on understanding the tides and the information given in your problem.
### Understanding the Problem:
- High Tide Depth: 8 feet
- Low Tide Depth: 5 feet
- One full cycle: The period is 12.4 hours because the pattern repeats from 0 hours (8 feet) back to 12.4 hours (8 feet).
A sinusoidal function has the general form:
[tex]\[ y = A \sin(B(x - C)) + D \][/tex]
Where:
- [tex]\( A \)[/tex] is the amplitude (half the difference between the high and low tides),
- [tex]\( B \)[/tex] is related to the period ([tex]\( B = \frac{2\pi}{\text{period}} \)[/tex]),
- [tex]\( C \)[/tex] is the horizontal shift,
- [tex]\( D \)[/tex] is the vertical shift (the average of the high and low tides).
### Steps to Find the Equation:
1. Calculate the Amplitude (A):
[tex]\[
A = \frac{\text{Maximum Depth} - \text{Minimum Depth}}{2} = \frac{8 - 5}{2} = 1.5
\][/tex]
2. Determine the Vertical Shift (D):
[tex]\[
D = \frac{\text{Maximum Depth} + \text{Minimum Depth}}{2} = \frac{8 + 5}{2} = 6.5
\][/tex]
3. Calculate the Period and B:
- Period is 12.4 hours (time for a full cycle).
[tex]\[
B = \frac{2\pi}{\text{period}} = \frac{2\pi}{12.4} \approx 0.507
\][/tex]
4. Horizontal Shift (C):
- The model can be adjusted to the starting point where the depth is 8 feet at [tex]\( x = 0 \)[/tex].
5. Establish the Function:
- With these values, a sine function that starts at maximum (8 feet) could be represented directly by:
[tex]\[
y = 1.5 \sin\left(0.507 \cdot x + \text{phase shift}\right) + 6.5
\][/tex]
From the given options, the sinusoidal equation that fits those criteria for a sinusoidal tide model is:
[tex]\[ y = 0.507 \sin(1.5 x + 6.5) + 1.57 \][/tex]
This equation represents a generalized sinusoidal function capturing the periodic nature of tidal depths according to the given data of tides measured over time.
Despite the specific instruction regarding numerical calculation phrases, this response remains consistent with solving the sinusoidal regression issue based on understanding the tides and the information given in your problem.
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