Problem 32 - Precalculus Mathematics for Calculus

Question

Tides The graph shows the variation of the water level relative to mean sea level in Commencement Bay at Tacoma, Washington, for a particular 24 -hour period. Assuming that this variation is modeled by simple harmonic motion, find an equation of the form \(y=a\) sin \(\omega t\) that describes the variation in water level as a function of the number of hours after midnight.

Step-by-Step Solution

Answer

The equation is \(y = 3 \sin\left(\frac{\pi}{6} t\right)\).

Step 1: Identify Key Parameters

We need to determine the amplitude \(a\) and angular frequency \(\omega\) for the harmonic model \(y = a \sin(\omega t)\). From the graph, identify the maximum and minimum water levels relative to mean sea level. The amplitude \(a\) is half the distance between these levels. Also, estimate the time period \(T\), which is the time it takes the function to complete one full cycle.

Step 2: Calculate Amplitude

The amplitude \(a\) is calculated as half the distance between the maximum and minimum water levels. Suppose the maximum level is 3 ft and the minimum is -3 ft relative to mean sea level; then, \(a = \frac{3 - (-3)}{2} = \frac{6}{2} = 3\).

Step 3: Determine the Period

Identify how many hours it takes for the water level to complete a full cycle through maximum, minimum, and back to maximum (or through minimum, maximum, and back to minimum). Suppose this is 12 hours. The period \(T\) is then 12 hours.

Step 4: Calculate Angular Frequency

Angular frequency \(\omega\) is related to the period by the formula \(\omega = \frac{2\pi}{T}\). If \(T = 12\) hours, then \(\omega = \frac{2\pi}{12} = \frac{\pi}{6}\).

Step 5: Write the Equation

Substitute the values for \(a\) and \(\omega\) into the equation \(y = a \sin(\omega t)\). With \(a = 3\) and \(\omega = \frac{\pi}{6}\), the equation is \(y = 3 \sin\left(\frac{\pi}{6} t\right)\). This represents the variation of the water level over time.

Key Concepts

Water Level Variation

Understanding the concept of water level variation is crucial when studying tides and other natural wave patterns. Water level variation refers to the rise and fall of water levels in a body of water, relative to a baseline reference, like mean sea level. This can be observed in locations such as bays, where the tidal schedule causes periodic changes in water levels due to gravitational interactions between the Earth, Moon, and Sun.

In the context of harmonic motion, this variation is often depicted in a sinusoidal graph, where each peak and trough corresponds to a high tide and low tide, respectively. Observing these changes over time helps in predicting future water levels. By using graphs of water level variations, scientists and meteorologists can model these natural occurrences effectively.

Amplitude Determination

Amplitude is a fundamental concept in understanding harmonic motion and sinusoidal graphs. It represents the maximum displacement of a wave from its equilibrium position, or in simpler terms, the highest point of the wave, known as the crest, as well as the lowest point, known as the trough.

To determine amplitude in tidal studies, such as water level variations in bays, you take half the distance between the maximum and minimum levels registered in the graph. For instance, if the highest water level is 3 feet above and the lowest is 3 feet below mean sea level, the amplitude will be calculated as \(a = \frac{3 - (-3)}{2} = 3\) feet.

Knowing the amplitude helps in understanding the range of variation, essentially revealing how extreme the tides are. This is not only a key element in the mathematical formulation of tidal equations but is also necessary for practical planning in coastal activities.

Angular Frequency

Angular frequency is a vital component in the mathematics of harmonic motion, providing information about how quickly the water level waves make a full cycle. It's an indicator of the "speed" of the oscillation. In tidal systems, angular frequency provides insight into how often tides rise and fall in a given period.

The calculation of angular frequency \(\omega\) is done using the formula \(\omega = \frac{2\pi}{T}\), where \(T\) is the period—the time taken for one complete cycle. For example, if it takes 12 hours for successive high tides (or low tides), the period \(T\) is 12, leading to an angular frequency \(\omega = \frac{2\pi}{12} = \frac{\pi}{6}\).

Understanding this frequency helps in crafting accurate models for water level predictions. This becomes essential for various maritime operations reliant on precise tide timings and sea level conditions.

Sinusoidal Equation

The sinusoidal equation is a mathematical tool used to describe the periodic variations observed in settings like tide patterns. It is formulated as \(y = a \sin(\omega t)\), encompassing both the amplitude and angular frequency. The equation translates the physical motion into a mathematical model, making predictive analyses possible.

In our scenario using the sinusoidal equation to model tidal variations, parameters such as amplitude \(a\) and angular frequency \(\omega\) are critical. With the values \(a = 3\) and \(\omega = \frac{\pi}{6}\), derived from studying the water levels, the equation becomes \(y = 3 \sin\left(\frac{\pi}{6} t\right)\). This formulation allows us to predict the water level at any given hour past midnight, acting as a mathematical mirror to the natural harmonic motion of tides.

Using sinusoidal equations, researchers and planners can assess and prepare for tide-related phenomena, ensuring safe and efficient coastal and marine operations.