Problem 31 - Algebra and Trigonometry

Question

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 ft above, drops to 21 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 tide height function is \( h(t) = 21\sin\left(\frac{\pi}{6}t\right) \).

Step 1: Understand the problem

The tide starts at mean sea level, rises to 21 feet above, then drops to 21 feet below sea level, and returns to mean sea level over a 12-hour period. This suggests that the tide motion is sinusoidal (simple harmonic motion). Our task is to express this as a mathematical function of time.

Step 2: Identify the characteristics of sinusoidal motion

In a simple harmonic motion, the maximum displacement from the mean position is known as the amplitude. For this problem, the amplitude is 21 feet, since the tide rises and falls 21 feet above and below mean sea level. The period of the motion is 12 hours.

Step 3: Formulate the equation

A general form for a sinusoidal function is: \[ h(t) = A imes ext{sin}(Bt + C) + D \]Where:- \( A \) is the amplitude (21 feet),- \( B \) affects the period of the function,- \( C \) is the phase shift (initial angle, set to 0 for simplicity),- \( D \) is the vertical shift (0 for mean sea level).The period \( T \) of a sine function is given by \( \frac{2\pi}{B} \). For this problem, \( T = 12 \) hours. Hence, \( B = \frac{2\pi}{12} = \frac{\pi}{6} \).

Step 4: Construct the equation

Using the values obtained, plug into the function:\[ h(t) = 21 imes ext{sin}\left(\frac{\pi}{6}t\right) \]This gives the tide height as a function of time \( t \).

Step 5: Sketch the graph

Plot the function \( h(t) = 21 imes ext{sin}\left(\frac{\pi}{6}t\right) \) over a 12-hour period. The graph starts at \( h = 0 \), reaches 21 feet at 3 hours, returns to 0 feet at 6 hours, falls to -21 feet at 9 hours, and returns to 0 feet at 12 hours, completing one full cycle.

Key Concepts

Sinusoidal Function

A sinusoidal function is a type of mathematical function that describes a smooth and continuous wave-like pattern. It is commonly used to model phenomena that exhibit periodic motion, like tides or sound waves. The basic form of a sinusoidal function can be written as: \[ h(t) = A \cdot \text{sin}(Bt + C) + D \] where:

• \( A \) is the amplitude, which represents the maximum vertical distance from the mean or equilibrium position.
• \( B \) determines the period or how quickly the function completes one cycle.
• \( C \) is the phase shift, affecting the horizontal position of the wave.
• \( D \) represents any vertical shifts that may move the baseline up or down.

These components together shape the behavior and position of the sinusoidal function. For the tides in the Bay of Fundy, this concept helps describe the height of the tides over time, showcasing a repetitive rise and fall motion.

Amplitude

Amplitude in the context of a sinusoidal function represents the maximum deviation of the function from its mean value. It is a measure of how large the waves are. In simple harmonic motion, it is the height from the equilibrium position to the peak of the wave.
For the tides at the Bay of Fundy, the amplitude is 21 feet. This means the water level rises 21 feet above the mean sea level and drops 21 feet below it. Unlike other features of the wave, the amplitude remains constant, assuming no changes in external conditions.
In general terms, the amplitude gives us an understanding of the energy of the motion; larger amplitudes indicate more energetic waves or tides.

Period of Motion

The period of motion in a sinusoidal function is the time it takes to complete one full cycle of the wave, from start to finish. It essentially tells how fast or slow the pattern repeats itself. In mathematical terms, the period \( T \) for a sine function is given by \( \frac{2\pi}{B} \), where \( B \) alters the frequency of the wave.
In our tide example, the period is 12 hours. This means it takes 12 hours to go from the mean sea level, reach the highest point, descend to the lowest point, and return to the starting level.
Understanding the period helps in predicting future behavior of the tide, showing precise intervals when high or low tides will occur in the Bay of Fundy.

Tide Cycle

A tide cycle involves regular changes in sea level caused by the gravitational forces exerted by the moon and sun. For locations like the Bay of Fundy, which experience substantial tidal ranges, these cycles are both fascinating and useful phenomena to study.
In terms of a sinusoidal model, a full tide cycle in this instance spans 12 hours. This means every 12-hour span encapsulates going from average sea level to the highest point, back through average, to the lowest point, and once again returning to average sea level.
Tide cycles dictate maritime activities, such as navigation, fishing, and construction projects, making it critical to predict these patterns accurately. The sinusoidal nature of tides means that they can often be calculated and forecasted using reliable mathematical models.