Question

The length of time between consecutive high tides is 12 hours and 25 minutes. According to the National Oceanic and Atmospheric Administration, on Saturday, April 21,2018 , in Sitka, Alaska, high tide occurred at 4: 51 AM (4.85 hours) and low tide occurred at 11:50 AM (11.83 hours). Water heights are measured as the amounts above or below the mean lower low water. The height of the water at high tide was 10.03 feet, and the height of the water at low tide was -0.46 feet. (a) Approximately when did the next high tide occur? (b) Find a sinusoidal function of the form $$ y=A \sin (\omega x-\phi)+B $$ that models the data. (c) Use the function found in part (b) to predict the height of the water at 3 PM.

Short answer

(a) Next high tide: 5:16 PM. (b) Function: \( y = 5.245 \sin (0.253 x + 0.3425) + 4.785 \). (c) Height at 3 PM: 0.374 feet.

Step-by-step solution

Calculate the time of the next high tide

The length of time between consecutive high tides is 12 hours and 25 minutes, which is 12.4167 hours. Add this duration to the initial high tide time (4.85 hours). \[ \text{Next High Tide Time} = 4.85 + 12.4167 = 17.2667 \] Convert 17.2667 hours into standard time: 17 hours and 0.2667*60 minutes. Thus, \( 17.2667 \text{ hrs} = 17 \text{ hrs} \ 16 \text{ mins} \). This translates to 5:16 PM.

Define the sinusoidal function for the tidal data

Given the amplitude, period, phase shift, and vertical shift, we infer: 1. The period is 2 times the time between consecutive high tides: \[ T = 2 \times 12.4167 = 24.8334 \text{ hours} \] The angular frequency \( \omega \) is computed as: \[ \omega = \frac{2 \pi}{T} = \frac{2 \pi}{24.8334} \approx 0.253 \] 2. The amplitude: \[ A = \frac{10.03 - (-0.46)}{2} = 5.245 \] 3. The vertical shift: \[ B = \frac{10.03 + (-0.46)}{2} = 4.785 \] 4. The phase shift:\( x = 4.85 \text{ hours} \) and this is when \( y = 10.03 \). Set up: \[ 10.03 = 5.245 \sin(0.253 \cdot 4.85 - \phi) + 4.785 \] Simplify to find \( \phi \): \[ 5.245 = 5.245 \sin(1.2281 - \phi) \] At \( x=4.85 \), \( \sin(1.2281 - \phi) = 1 \), hence \( 1.2281 - \phi = \frac{\pi}{2} \), which gives \( \phi \approx 1.2281 - \frac{\pi}{2} \approx -0.3425 \). Thus, \[ y = 5.245 \sin (0.253 x + 0.3425) + 4.785 \].

Predict the water height at 3 PM

Convert 3 PM to hours since midnight: \( 15 \text{ hours} \). Using the function: \[ y = 5.245 \sin (0.253 \cdot 15 + 0.3425) + 4.785 \] Compute: \[ y = 5.245 \sin (4.1375) + 4.785 \] Evaluate \( \sin(4.1375) \approx -0.841 \), thus: \[ y = 5.245 (-0.841) + 4.785 \approx -4.411 + 4.785 = 0.374 \text{ feet} \].

Key concepts

Tidal Patterns

Tidal patterns refer to the predictable rise and fall of sea levels caused by the gravitational forces exerted by the moon, the sun, and the rotation of the Earth. In coastal regions like Sitka, Alaska, understanding tidal patterns is essential for navigation, fishing, and various marine activities.
Tides follow a regular cycle, with high and low tides repeating approximately every 12 hours and 25 minutes. This is known as a semi-diurnal pattern, meaning there are usually two high tides and two low tides each day.
High tides occur when the ocean's water levels are at their peak, while low tides happen when the water is at its lowest. These changes in water levels are measured relative to a baseline known as the mean lower low water (MLLW). For instance, if the height of the water at high tide is 10.03 feet above MLLW and the low tide is -0.46 feet below MLLW, we'd use these values to model the tidal changes mathematically.

Trigonometric Modeling

Trigonometric modeling involves using trigonometric functions, such as sine and cosine, to represent periodic phenomena like tidal patterns. The basic form of a sinusoidal (sine-based) function is:
\[ y = A \sin(\omega x - \phi) + B \]
Here are the components of this function:

• \textbf{A} is the amplitude, indicating the peak height of the wave.
• \textbf{\textbackslash omega} is the angular frequency, which is related to the period of the wave.
• \textbf{\textbackslash phi} is the phase shift, which determines the horizontal shift of the wave.
• \textbf{B} is the vertical shift, changing the centerline of the wave up or down.

To model tidal data specifically, we base our function on measured high and low tide values. We need to calculate the amplitude and period of the tides. Once these values are determined, we apply them to construct our sinusoidal model, allowing us to predict future tides and water heights accurately.

Amplitude and Period

The amplitude and period are crucial elements in trigonometric modeling of tidal patterns. They define the height and the time interval of the repeating waves.

• \textbf{Amplitude (A)}: This represents half the difference between the maximum and minimum values of the wave. In our tidal example, the maximum height is 10.03 feet, and the minimum is -0.46 feet. Thus, the amplitude is calculated as:
  \[ A = \frac{10.03 - (-0.46)}{2} = 5.245 \] This means the water level varies 5.245 feet above and below the average level.

• \textbf{Period (T)}: The period is the total time it takes for the wave to complete one full cycle. For tides, we observe the time between consecutive high tides and double this value to account for both the high and low tides within a cycle. Given the period between high tides is 12 hours and 25 minutes, equivalent to 12.4167 hours, the full period is:
  \[ T = 2 \times 12.4167 = 24.8334 \text{ hours} \] The period tells us that a full tidal cycle of rise and fall repeats approximately every 24 hours and 50 minutes.

Using these two parameters, we can determine the angular frequency as:
\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{24.8334} \approx 0.253 \]
This angular frequency is crucial for accurately modeling the wave's cycles.