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 Charleston, South Carolina, high tide occurred at 12:25 AM (0.42 hours) and low tide occurred at 7: 05 à \((7.08\) 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 5.88 feet, and the height of the water at low tide was 0.06 foot. (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 \(5 \mathrm{PM}\).

Short answer

The next high tide occurs at 12:50 PM. The function is \(y = 2.91 \sin(0.506x - 0.21252 + \frac{\pi}{2}) + 2.97\). The water height at 5 PM is approximately 4.403 feet.

Step-by-step solution

Calculate the Next High Tide Time

The length of time between consecutive high tides is 12 hours and 25 minutes. Adding this to the previous high tide at 12:25 AM (0.42 hours from midnight), we get:0.42 hours + 12 hours 25 minutes = 0.42 + 12 + 25/60 = 0.42 + 12 + 0.417 ≈ 12.837 hours.So, the next high tide will occur at approximately 12.837 hours from midnight, which is around 12:50 PM.

Determine the Amplitude (A)

The amplitude is half the difference between the maximum and minimum water heights. Here, the heights are 5.88 feet for high tide and 0.06 feet for low tide. Thus:\[ A = \frac{5.88 - 0.06}{2} = \frac{5.82}{2} = 2.91 \text{ feet} \]

Find the Vertical Shift (B)

The vertical shift is the average of the maximum and minimum water heights:\[ B = \frac{5.88 + 0.06}{2} = \frac{5.94}{2} = 2.97 \text{ feet} \]

Determine the Angular Frequency (ω)

One complete period, which is the time between consecutive high tides, is 12 hours 25 minutes (12.417 hours). The angular frequency is given by:\[ \omega = \frac{2\pi}{T} = \frac{2\pi}{12.417} \approx 0.506 \text{ radians/hour} \]

Determine the Phase Shift (φ)

At high tide (0.42 hours), let x = 0.42 in hours. Since the function is at its maximum (A + B) at high tide, we can solve for φ when \(\sin(\omega x - \phi) = 1\):\( 0.42 \omega - \phi = \frac{\pi}{2} \)\( 0.506 \times 0.42 - \phi = \frac{\pi}{2} \)\( 0.21252 - \phi = \frac{\pi}{2} \)\( \phi = 0.21252 - \frac{\pi}{2} \)

Write the Sinusoidal Function

Now that we have A, B, ω, and φ, the sinusoidal function is:\[ y = 2.91 \sin(0.506x - 0.21252 + \frac{\pi}{2}) + 2.97 \]

Predict the Height of the Water at 5 PM

Convert 5 PM to hours from midnight: 5 PM is 17 hours from midnight. Substitute x = 17 into the function:\( y = 2.91 \sin(0.506 \times 17 - 0.21252 + \frac{\pi}{2}) + 2.97 \)Calculate inside the sine function first:\( 0.506 \times 17 - 0.21252 \approx 8.392 \)Now add \(\frac{\pi}{2}\):\( 8.392 + \frac{\pi}{2} \approx 8.392 + 1.5708 \approx 9.9628 \)Thus:\( y = 2.91 \sin(9.9628) + 2.97 \)Approximating \( \sin(9.9628) \approx 0.49 \):\( y = 2.91 \times 0.49 + 2.97 \approx 4.403 \text{ feet} \)

Key concepts

Sinusoidal Functions

Understanding sinusoidal functions is essential for modeling phenomena with periodic behavior, like tides. Sinusoidal functions come in the form: $$ y = A \sin(\omega x - \phi) + B $$ Here, we have several components:

• A (Amplitude): Determines the height of the wave's peaks.
• \omega (Angular Frequency): Determines how many cycles occur in a given time.
• \phi (Phase Shift): Adjusts where the wave starts.
• B (Vertical Shift): Moves the wave up or down.

These functions are incredibly versatile and help us model natural occurrences such as daylight hours, temperature seasons, and even tides effectively.

Tide Prediction

Tide prediction involves using trigonometric patterns to forecast the height and timing of tides. The tides follow a regular, predictable pattern governed by lunar cycles, making sinusoidal functions perfect for this task. To predict the next high tide given a cycle length of 12 hours and 25 minutes, you add this duration to the time of the last high tide. For example, starting from 12:25 AM, the next high tide would be around 12:50 PM. This predictable periodicity allows you to plan activities such as fishing and coastal navigation more efficiently.

Trigonometric Modeling

Trigonometric modeling involves creating mathematical models to describe cycles and oscillations, like tides. This technique uses sinusoidal functions due to their periodic nature. For the case of tide heights, we collect data on times and tide heights, then fit them into a trigonometric function. Here's the step-by-step process for trigonometric modeling:

• Determine the amplitude (the height difference between high and low tide).
• Calculate the angular frequency (how fast the cycle repeats).
• Find the phase shift (when the cycle starts).
• Identify the vertical shift (average tide height).

Once the parameters are known, they help predict future values like when the tides will be and how high they will reach.

Amplitude Calculation

Amplitude is a crucial concept in understanding the extent of variation in tides. It represents half the distance between the highest and lowest points of the tide. Calculating amplitude involves finding the maximum and minimum tide heights and using the formula: $$ A = \frac{\text{Max Height} - \text{Min Height}}{2} $$ In our example:

• High tide: 5.88 feet
• Low tide: 0.06 feet
• Amplitude: $$ A = \frac{5.88 - 0.06}{2} = 2.91 \text{ feet} $$

Amplitude helps us understand the effectiveness of tide height over the period, giving insight into coastal activities and safety.

Angular Frequency

Angular frequency indicates how many cycles occur within a given period and is crucial for identifying the frequency of tidal cycles. It's calculated by the formula: $$ \omega = \frac{2\pi}{T} $$ where T is the period (time between consecutive high tides). For example, if the period is 12 hours and 25 minutes (12.417 hours), then: $$ \omega = \frac{2\pi}{12.417} \approx 0.506 \text{ radians/hour} $$ Angular frequency allows us to understand how rapidly the tidal cycles repeat and is essential for accurate tide predictions and understanding the dynamics of coastal environments.