Problem 49 - Precalculus: Functions and Graphs

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 amplitude 0.5, period 12 hours, shifted 5.5 hours right.

Step 1: Identify Function Parameters

The given function is \( f(t) = a \cos \left(\frac{\pi}{6} t - \frac{11\pi}{12}\right) \). We are given \( a = 0.5 \). Thus, the function simplifies to: \[ f(t) = 0.5 \cos \left(\frac{\pi}{6} t - \frac{11\pi}{12}\right) \] Here, \( a = 0.5 \) is the amplitude, \( b = \frac{\pi}{6} \) determines the frequency, and \( c = -\frac{11\pi}{12} \) is the phase shift.

Step 2: Determine Amplitude and Period

The amplitude of the function, given by \( a \), is 0.5 meters, representing the maximum height of the tide deviation from the mean level.To find the function's period, use: \[ \, \text{Period} = \frac{2\pi}{b} = \frac{2\pi}{\frac{\pi}{6}} = 12 \, \text{hours} \] This indicates the function repeats every 12 hours.

Step 3: Determine Phase Shift

The phase shift \( c \) can be calculated using the formula: \[ \text{Phase shift} = -\frac{c}{b} = -\frac{-\frac{11\pi}{12}}{\frac{\pi}{6}} = \frac{66}{12} = 5.5 \] This means the graph is shifted 5.5 hours to the right.

Step 4: Sketch the Graph

1. Draw the horizontal axis (time, \( t \), in hours) and the vertical axis (height in meters).2. Since the period is 12 hours, mark the x-axis from 0 to 12 and continue repeating every 12 hours.3. The amplitude is 0.5 meters, so the y-coordinates will range from -0.5 to 0.5.4. Start the graph at 5.5 on the x-axis due to the phase shift, starting from 0.5 at its peak.5. The wave will then return to 0, decrease to -0.5 (trough), return to 0, and back to its peak over the span of the 12-hour period.

Key Concepts

Tidal Components

Tidal components are important in understanding the behavior of tides, which can be modeled using trigonometric functions. A tidal component represents the factors that affect the height of the tide at any given point along the shore. These components include various forces like the gravitational pull from the moon and the sun, which cause water to rise and fall periodically.

To model these tides mathematically, we use trigonometric functions such as sine and cosine. These functions help predict how the tide’s height changes over time. In the exercise, the function given is structured as:

• \( f(t) = a \cos(b t + c) \)
• where \(a\) represents the amplitude, \(b\) relates to the frequency, and \(c\) modifies the phase shift.

Understanding these components is essential for accurately predicting tide behaviors, which is crucial for navigation, fishing, and coastal activities.

Amplitude and Period

Amplitude and period are key parameters in understanding trigonometric functions used for modeling phenomena like tides. The amplitude of a function describes the extent of the wave's deviation from its central axis.

In our function, the amplitude is given as \(a = 0.5 \) meters. This means the maximum height that the tide will rise above or drop below the mean water level is 0.5 meters.

• Amplitude translates visually to the height of the wave's crest or the depth of its trough.

The period of the function is the duration it takes for one complete cycle of the wave to occur. For the function described as:

• \( f(t) = 0.5 \cos \left(\frac{\pi}{6} t - \frac{11\pi}{12}\right) \)
• The period is computed with \(\text{Period} = \frac{2\pi}{b} = 12\) hours.

This twelve-hour period indicates the time interval in which the tide cycle repeats itself, meaning you would observe similar tide heights every 12-hour span.

Phase Shift

Phase shift in a trigonometric function refers to the horizontal shift of the graph along the time axis. It determines where the tidal cycle begins in relation to the expected time.

In the given equation,

• \( f(t) = 0.5 \cos \left(\frac{\pi}{6} t - \frac{11\pi}{12}\right) \)
• the phase shift is calculated as \( \frac{66}{12} \) which equates to a 5.5-hour shift to the right.

This simply means the expected high tide is delayed by 5.5 hours. If midnight is set as \(t=0\), the position of the cycles has been adjusted such that they peak or trough 5.5 hours later than they would have without this phase shift.

Understanding phase shift is crucial, especially for precise planning of activities that depend on tidal conditions, like docking ships or scheduling surfing sessions.