Question
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 t = 0 be 12:00 am.
Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.
Which periodic function, sine or cosine, would be a simpler model for the situation? Explain.
Answers
GPT-4o mini
To determine whether a sine or cosine function would be a simpler model for the periodic changes in tide depth, let's first examine the given information:
1. **Tide Information**:
- **Low Tides:**
- 12:00 am (t = 0), depth = 2.5 m
- 12:30 pm (t = 12.5), depth = 2.5 m
- **High Tides:**
- 6:15 am (t = 6.25), depth = 5.5 m
- 6:45 pm (t = 18.75), depth = 5.5 m
2. **Periodicity**:
- The transition from low tide to high tide suggests a periodic wave function likely to be modeled as either sine or cosine.
3. **Characteristics of Sine and Cosine**:
- The cosine function typically starts at its maximum value (A) when t = 0, while the sine function starts at its equilibrium position (0).
- Given that the low tide occurs at t = 0, it suggests that we have a minimum value at the starting point.
### Analysis
- Since the lowest point (low tide) is observed at t = 0 (which corresponds to 12:00 am), and also at t = 12.5 hours (12:30 pm), this aligns perfectly with the behavior of a cosine function. A cosine function can be adjusted to start at a minimum if we shift it appropriately.
- Furthermore, the high tide occurs exactly halfway (in time) between low tides, suggesting a standard wave shape, which is characteristic of a cosine function when appropriately adjusted:
- The depth varies between 2.5 m (low tide) and 5.5 m (high tide), thus:
- Amplitude \( A = (5.5 - 2.5) / 2 = 1.5 \, m \)
- Midline (equilibrium) \( D = (5.5 + 2.5) / 2 = 4.0 \, m \)
### Conclusion
The cosine function, with an appropriate phase shift, would model this situation effectively because:
1. It can start at a minimum value when adjusted appropriately.
2. The period aligns well with the observed pattern of tides.
3. The amplitude and midline can easily be derived from the given depths.
An example of the cosine function model might look like this:
\[ y = 1.5 \cos\left(\frac{2\pi}{12}\left(t - 6.25\right)\right) + 4.0 \]
This function captures the periodic tidal changes clearly, confirming that cosine would be a simpler choice for modeling this tide situation.
1. **Tide Information**:
- **Low Tides:**
- 12:00 am (t = 0), depth = 2.5 m
- 12:30 pm (t = 12.5), depth = 2.5 m
- **High Tides:**
- 6:15 am (t = 6.25), depth = 5.5 m
- 6:45 pm (t = 18.75), depth = 5.5 m
2. **Periodicity**:
- The transition from low tide to high tide suggests a periodic wave function likely to be modeled as either sine or cosine.
3. **Characteristics of Sine and Cosine**:
- The cosine function typically starts at its maximum value (A) when t = 0, while the sine function starts at its equilibrium position (0).
- Given that the low tide occurs at t = 0, it suggests that we have a minimum value at the starting point.
### Analysis
- Since the lowest point (low tide) is observed at t = 0 (which corresponds to 12:00 am), and also at t = 12.5 hours (12:30 pm), this aligns perfectly with the behavior of a cosine function. A cosine function can be adjusted to start at a minimum if we shift it appropriately.
- Furthermore, the high tide occurs exactly halfway (in time) between low tides, suggesting a standard wave shape, which is characteristic of a cosine function when appropriately adjusted:
- The depth varies between 2.5 m (low tide) and 5.5 m (high tide), thus:
- Amplitude \( A = (5.5 - 2.5) / 2 = 1.5 \, m \)
- Midline (equilibrium) \( D = (5.5 + 2.5) / 2 = 4.0 \, m \)
### Conclusion
The cosine function, with an appropriate phase shift, would model this situation effectively because:
1. It can start at a minimum value when adjusted appropriately.
2. The period aligns well with the observed pattern of tides.
3. The amplitude and midline can easily be derived from the given depths.
An example of the cosine function model might look like this:
\[ y = 1.5 \cos\left(\frac{2\pi}{12}\left(t - 6.25\right)\right) + 4.0 \]
This function captures the periodic tidal changes clearly, confirming that cosine would be a simpler choice for modeling this tide situation.