Problem 33 - Precalculus Mathematics for Calculus

Question

Tides The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12 -hour period the water starts at mean sea level, rises to \(21 \mathrm{ft}\) above, drops to \(21 \mathrm{ft}\) below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12 -hour period.

Step-by-Step Solution

Answer

The equation is \( h(t) = 21 \cos\left(\frac{\pi}{6} t\right) \). The graph is a cosine wave over 12 hours.

Step 1: Identify the Properties of Simple Harmonic Motion

The motion of tides is assumed to be simple harmonic, which means it can be modeled using sine or cosine functions. Key components include the amplitude (distance from mean to peak), the period (time for a full cycle), and the phase shift (horizontal shift along time axis).

Step 2: Determine the Amplitude

The amplitude is the maximum deviation from the mean sea level, which is given as 21 feet. Thus, the amplitude is 21 feet.

Step 3: Determine the Period

The period is the time it takes for the tide to complete one full cycle, rising from the mean level, reaching the peak, descending to the trough, and returning to the mean. This is given as 12 hours.

Step 4: Write the Equation for Simple Harmonic Motion

The general form of a simple harmonic motion equation is \[ h(t) = A \cos\left(\frac{2\pi}{T} t + \phi\right) \] where - \( A \) is the amplitude, here \(21\) feet,- \( T \) is the period, here \(12\) hours, - \( \phi \) is the phase shift, which is zero if we assume that the tide starts at the mean sea level following a cosine curve.The equation becomes: \[ h(t) = 21 \cos\left(\frac{2\pi}{12} t\right) \]

Step 5: Verify and Interpret the Equation

Check the equation by plugging in different times (e.g., \( t = 0, 3, 6, 9, 12 \)) to ensure that the tide height starts at \(0\), rises to \(21\) feet, descends to \(-21\) feet, and returns to zero over a 12-hour period.

Step 6: Sketch the Graph

Plot the function \( h(t) = 21 \cos\left(\frac{2\pi}{12} t\right) \) over the time interval from 0 to 12 hours. The graph should show a cosine wave starting from (0,0), peaking at (3,21), crossing back through (6,0), reaching a trough at (9,-21), and returning to (12,0).

Key Concepts

Amplitude

In simple harmonic motion, amplitude is a crucial concept. It is the maximum distance that the wave (in this case, the height of the tide) reaches from its average position or the mean sea level. For the Bay of Fundy, the amplitude is determined by the maximum height the tide reaches, which is given as 21 feet.

Consider amplitude as the indicator of how "strong" or "tall" the wave is. It changes how far up and down the wave can go. The larger the amplitude, the more extreme the high and low points.

In a mathematical model, amplitude is represented by the variable \( A \). For this particular tidal model in the Bay of Fundy, the wave's maximum positive and negative reach from the mean sea level determines the amplitude. Thus, the equation becomes:

• \( A = 21 \) feet

By understanding amplitude, you can predict the highest and lowest points the tide can reach based on the simple harmonic model.

Period

The period is another fundamental attribute of simple harmonic motion. It refers to the time taken for the wave to complete one full cycle. For the Bay of Fundy's tides, this means the time from one high tide to the next or from mean sea level until the tide returns to that level.

The period is directly related to the frequency, which is how often the waves repeat over time. In this model, the tide completes a cycle every 12 hours, which is noted as the period \( T \).

To incorporate the period in the mathematical function, use the formula:

• \( T = 12 \) hours

This means every 12 hours, the tide follows a predictable path: starting from the mean sea level, rising to the peak, dipping down to the lowest point, and coming back to the starting level.

Period is vital for understanding when the tide reaches its peak and trough throughout the day.

Graphing Trigonometric Functions

Visualizing simple harmonic motion through graphing is a powerful way to understand how physical phenomena behave over time. When graphing the tides, you'd use a trigonometric function, here a cosine function, to represent the periodic movement.

A trigonometric graph starts at the mean position and moves through a series of peaks and troughs. The wave pattern offers insightful views into the motion's characteristics, such as how long it takes to reach a peak or how low it dips.

For the Bay of Fundy, we use the cosine function because it begins at the maximum point when there is no phase shift. Graphically depicting the equation:

• \( h(t) = 21 \cos\left(\frac{2\pi}{12} t\right) \)

The graph starts at zero at \( t = 0 \), shows a peak at 3 hours, crosses back through the central axis at 6 hours, bottoms out at 9 hours, and comes back at 12 hours. This graphical representation helps to visually confirm the periodic and oscillating nature of the simple harmonic motion. The peaks and troughs represent the high and low tides, respectively.