Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

The height of the tide at a particular point on shore can be predicted by using seven trigonometric functions (called tidal components) of the form $$ f(t)=a \cos (b t+c) . $$ The principal lunar component may be approximated by $$ f(t)=a \cos \left(\frac{\pi}{6} t-\frac{11 \pi}{12}\right), $$ where \(t\) is in hours and \(t=0\) corresponds to midnight. Sketch the graph of \(f\) if \(a=0.5 \mathrm{~m}\).

Step-by-Step Solution

Answer

The graph is a cosine wave with a 0.5 m amplitude, 12-hour period, starting at 5.5 hours after midnight.

Step 1: Identify the Function

The given trigonometric function is: \[ f(t) = 0.5 \cos \left( \frac{\pi}{6} t - \frac{11\pi}{12} \right) \] This function will be graphed over a period in terms of hours.

Step 2: Determine the Period of the Function

To find the period of the cosine function, use the formula \( T = \frac{2\pi}{b} \), where \( b = \frac{\pi}{6} \).This gives:\[ T = \frac{2\pi}{\frac{\pi}{6}} = 12 \text{ hours} \] This means the function repeats its pattern every 12 hours.

Step 3: Identify the Phase Shift

The phase shift is given by \( -\frac{c}{b} \). Here, \( c = -\frac{11\pi}{12} \) and \( b = \frac{\pi}{6} \).Calculate:\[ \text{Phase Shift} = \frac{\frac{11\pi}{12}}{\frac{\pi}{6}} = \frac{11}{2} = 5.5 \text{ hours} \] The graph is shifted 5.5 hours to the right.

Step 4: Find the Amplitude

The amplitude of the function is the absolute value of \( a \), which is given as \( 0.5 \text{ m} \).Amplitude determines the vertical stretch of the graph and indicates that the graph reaches 0.5 meters above and below its center line.

Step 5: Plot Key Points and Sketch the Graph

Start plotting the graph by marking key points, considering the phase shift, amplitude, and period:- At \( t = 5.5 \), begin at the maximum value of \( 0.5\).- At \( t = 5.5 + 3 = 8.5 \), it crosses the center line.- At \( t = 5.5 + 6 = 11.5 \), reaches the minimum value of \( -0.5\).- At \( t = 5.5 + 9 = 14.5 \), it crosses the center line again.- At \( t = 5.5 + 12 = 17.5 \), completes the period back at maximum.The resulting graph oscillates between \(-0.5\) m and \(0.5\) m, completing one cycle every 12 hours.

Key Concepts

Tide Prediction

Understanding tide prediction is crucial for coastal activities and ecology. Tides are the periodic rise and fall of sea levels caused by gravitational interactions between Earth, the Moon, and the Sun. Predicting tides accurately helps in variety of applications such as fishing, shipping, and coastal management.
In the context of trigonometry, tidal movements are modeled using trigonometric functions. In particular, the cosine function is often used because it smoothly oscillates, which fits the regular and predictable nature of tides. By analyzing these mathematical models, we can forecast the timing and height of these oceanic movements, allowing for better planning and safety measures.
This is why, in exercises like the one given, trigonometric functions often play a key role in simulating the cycling behaviors of tides using specific parameters like amplitude, period, and phase shift.

Cosine Function

The cosine function is a fundamental trigonometric function that is periodically oscillating. In this model, the function is represented as: \[ f(t) = a \cos \left( \frac{\pi}{6} t - \frac{11\pi}{12} \right) \]This expression gives the height of the tide at any time, t, by using the cosine wave's properties.
The cosine function is suitable here because of its natural periodicity. It inherently repeats its pattern at regular intervals, perfectly mirroring the regularity of tides due to its inherent cyclical nature.

• The argument inside the cosine function: \( \frac{\pi}{6} t - \frac{11\pi}{12} \) governs how quickly the function oscillates (period) and where the oscillations begin on the time axis (phase shift).
• The parameter \( a \) determines how extreme these oscillations get, known as the amplitude.

This correlation between astronomical bodies and tidal patterns can thus be elegantly captured and studied using the cosine function.

Amplitude

Amplitude in the context of trigonometric functions is the measure of how far the graph of the function rises and falls from its central axis. Specifically, in the tide prediction exercise, the amplitude is provided as 0.5 meters.
Amplitude affects the magnitude of oscillation:

• The tide will rise to 0.5 m above and sink to 0.5 m below its average sea level, showcasing the total range of 1 m between the maximum and minimum tide heights.

Amid the peaks and troughs of the tide, the amplitude quantifies the vertical stretch of the cosine wave graph, essentially indicating how energetic the tidal movements are. It’s crucial for understanding not just the height of the tides, but the possible impact on coastlines and maritime operations.

Period

The period of a function in trigonometric context describes how long it takes for the function to complete one full cycle of its behavior. For the cosine function used here, the period is particularly calculated by finding:\[ T = \frac{2\pi}{b} \]where \( b = \frac{\pi}{6} \), yielding a period of 12 hours.
This mathematical concept translates into natural phenomena like the repeating tide cycles. Here, it indicates that the tide will go through its complete series of highs and lows within 12-hour intervals. Therefore, understanding the period helps forecast when high and low tides will occur on any given day.
Utilizing the period, predictions can be mapped about high or low tides at specific times, aiding in navigation and safety planning for maritime activities.

Phase Shift

Phase shift refers to the horizontal movement of a graph on the Cartesian plane. In this exercise, the phase shift is determined by the formula:\[ \text{Phase Shift} = \frac{-c}{b} \]where \( c = -\frac{11\pi}{12} \) and \( b = \frac{\pi}{6} \), resulting in a shift of 5.5 hours to the right.
This means the peak of the tides does not start exactly at the 0-hour mark but occurs after a 5.5-hour delay from this starting point. This offset is critical for predicting the exact timing when high tides or low tides will occur.
Comprehending phase shifts is essential when aligning mathematical predictions with real-world observations. It ensures that tide forecasts are accurate, matching them to expected occurrences instead of merely periodic intervals without real-world correlation.